fundamental of college geometry

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FundamentalsofCOLLEGEGEOMET~SECONDEDITIONIIII.EdwinM.HemmerlingDepartmentofMathematicsBakersfieldCollegeJOHNWILEY&SONS,NewYork8Chichisterl8Brisbane8TorontoI PrefaceBeforerevlsmgFundamentalsofCollegeGeometry,extensivequestionnairesrcresenttousersoftheearlieredition.Aconsciousefforthasbeenmadeinthiseditiontoincorporatethemanyfinesuggestionsgiventherespondentstothequestionnaire.Atthesametime,Ihaveattemptedtopreservethefeaturesthatmadetheearliereditionsopopular.Thepostulationalstructureofthetexthasbeenstrengthened.Somedefinitionshavebeenimproved,makingpossiblegreaterrigorinthedevelop-mentofthetheorems.Particularstresshasbeencontinuedinobservingthedistinctionbetweenequalityandcongruence.Symbolsusedforsegments,intervals,rays,andhalf-lineshavebeenchangedinorderthatthesymbolsforthemorecommonsegmentandraywillbeeasiertowrite.However,asymbolfortheintervalandhalf-lineisintroduced,whichwillstilllogicallyshowtheir,relationstothesegmentandray,Fundamentalspaceconceptsareintroducedthroughoutthetextinordertopreservecontinuity.However,thepostulatesandtheoremsonspacegeometryarekepttoaminimumuntilChapter14.Inthischapter,partic-ularattentionisgiventomensurationproblemsdealingwithgeometricsolids.GreateremphasishasbeenplacedonutilizingtheprinciplesofdeductivelogiccoveredinChapter2inderivinggeometrictruthsinsubsequentchapters.Venndiagramsandtruthtableshavebeenexpandedatanumberofpointsthroughoutthetext.thereisawidevanancethroughouttheUl1ltedStatesinthetimespentingeometryclasses,Approximatelytwofifthsoftheclassesmeetthreedaysaweek.Anothertwofifthsmeetfivedayseachweek,Thestudentwhostudiesthefirstninechaptersofthistextwillhavecompletedawell-roundedCopyright@1970,byJohnWiley&Sons,Ine.minimumcourse,includingallofthefundamentalconceptsofplaneandAllrightsreserved.spacegeometry.Eachsubsequentchapterinthebookiswrittenasacompletepackage,Reproductionortranslationofanypartofthisworkbeyondthatnoneofwhichisessentialtothestudyofanyoftheotherlastfivechapters,permittedbySections107or108ofthe1976UnitedStatesCopy-rightActwithoutthepermissionofthecopyrightownerisunlaw-veteachwillbroadenthetotalbackgroundofthestudent.Thiswillpermitful.Requestsforpermissionorfurtherinformationshouldbetheinstructorconsiderablelatitudeinadjustinghiscoursetothetimeavail-Wiley&Sons,Inc.addressedtothePermissionsDepartment,J.ohnableandtotheneedsofhisstudents.2019181716151413Eachchaptercontainsseveralsetsofsummarytests.ThesevaryintypetoLibraryofCongressCatalogueCardNumber:75-82969includetrue-falsetests,completiontests,problemstests,andproofstests.Akeyforthesetestsandtheproblemsetsthroughoutthetextisavailable.SBN47]370347PrintedintheUnitedStatesofAmericaJanuarv1969EdwinM.Hemmerlingv .PrefacetoFirstEditionDuringthepastdecadetheentireapproachtotheteachingofgeometryhasbccnundergoingseriousstudybyvariousnationallyrecognizedprofessionalgroups.Thisbookreflectsmanyoftheirrecommendations.ThestyleandobjectivesofthisbookarethesameasthoseofmyCollegePlaneGeometry,outofwhichithasgrown.BecauseIhaveaddedasignifi-cantamountofnewmaterial,however,andhaveincreasedtherigorem-ployed,ithasseemeddesirabletogivethebookanewtitle.InFunda-mentalsofCollegeGeometry,thepresentationofthesu~jecthasbeenstrength-encdbytheearlyintroductionandcontinueduseofthelanguageandsymbolismofsetsasaunifyingconcept.Thisbookisdesignedforasemester'swork.Thestudentisintroducedtothebasicstructureofgeometryandispreparedtorelateittoeverydayexperienceaswellastosubsequentstudyofmathematics.Thevalueofthepreciseuseoflanguageinstatingdefinitionsandhypo-thesesandindevelopingproofsisdemonstrated.Thestudentishelpedtoacquireanunderstandingofdeductivethinkingandaskillinapplyingittomathematicalsituations.Heisalsogivenexperienceintheuseofinduc-tion,analogy,andindirectmethodsofreasoning.Abstractmaterialsofgeometryarerelatedtoexperiencesofdailylifeofthestudent.Helearnstosearchforundefinedtermsandaxiomsinsuchareasofthinkingaspolitics,sociology,andadvertising.Examplesofcircularreasoningarestudied.Inadditiontoprovidingforthepromotionofproperattitudes,under-standings,andappreciations,thebookaidsthestudentinlearningtobecriticalinhislistening,reading,andthinking.Heistaughtnottoacceptstatementsblindlybuttothinkclearlybeforeformingconclusions.Thechapteroncoordinategeometryrelatesgeometryandalgebra.Propertiesofgeometricfiguresarethendeterminedanalyticallywiththeaidofalgebraandtheconceptofone-to-onecorrespondence.Ashortchapterontrigonometryisgiventorelateratio,similarpolygons,andcoordinategeometry.Illustrativeexampleswhichaidinsolvingsubsequentexercisesareusedliberallythroughoutthebook.Thestudentisabletolearnagreatdealofthematerialwithouttheassistanceofaninstructor.Throughoutthebookheisaffordedfrequentopportunitiesfororiginalandcreativethinking.Manyofthegeneroussupplyofexercisesincludedevelopmentswhichpreparefortheoremsthatappearlaterinthetext.Thestudentisledtodiscoverforhimselfproofsthatfollow.VII .ContentsThesummarytestsplacedattheendofthebookincludecompletion,true-false,multiple-choiceitems,andproblems.Theyaffordthestudentandtheinstructorareadymeansofmeasuringprogressinthecourse.EdwinM.HemmerlingI.BasicElementsofGeometryBakersfield,California,19642.ElementaryLogi.c513.DeductiveReasoning724.Congruence-CongruentTriangles1015.ParallelandPerpendicularLines1396.Polygons-Parallelograms1837.Circles2068.Proportion-SimilarPolygons2459.Inequalities28310.GeometricConstructions303II.GeometricLoci31912.AreasofPolygons34013.CoordinateGeometry36014.AreasandVolumesofSolids388------------------.------------------Appendix417GreekAlphabet419SymbolsandAbbreviations419Table1.SquareRoots421PropertiesofRealNumberSystem422ListofPostulates423ListsofTheoremsandCorollaries425AnswerstoExercises437Index459ixVlll 111BasicElementsofGeometry1.1.Historicalbackgroundofgeometry.Geometryisastudyofthepro-pertiesandmeasurementsoffigurescomposedofpointsandlines.Itisaveryoldscienceandgrewoutoftheneedsofthepeople.Thewordgeo-metryisderivedfromtheGreekwordsgeo,meaning"earth,"andmetrein,meaning"tomeasure."TheearlyEgyptiansandBabylonians(4000-3000E.C.)wereabletodevelopacollectionofpracticalrulesformeasuringsimplegeometricfiguresandfordeterminingtheirproperties.Theseruleswereobtainedinductivelyoveraperiodofcenturiesoftrial<111(1error.Theywerenotsupportedbyanyevidenceoflogicalproof.ApplicationsoftheseprincipleswerefoundinthebuildingofthePyramidsandthegreatSphinx.TheirrigationsystemsdevisedbytheearlyEgyptiansindicatethattheyhadanadequateknowledgeofgeometryasitmaybeappliedinlandsurveying.TheBabylonianswereusinggeometricfiguresintiles,walls,anddecorationsoftheirtemples.FromEgyptandBabyloniatheknowledgeofgeometrywastakentoGreece.FromtheGreekpeoplewehavegainedsomeofthegreatestcon-tributionstotheadvancementofmathematics.TheGreekphilosophersstudiedgeometrynotonlyforutilitarianbenefitsderivedbutfortheestheticandculturaladvantagesgained.TheearlyGreeksthrivedonaprosperousseatrade.Thisseatradebroughtthemnotonlywealthbutalsoknowledgefromotherlands.ThesewealthycitizensofGreecehadconsiderabletimeforfashionabledebatesandstudyonvarioustopicsofculturalinterestbe-causetheyhadslavestodomostoftheirroutinework.UsuallytheoriesandconceptsbroughtbackbyreturningseafarersfromforeignlandsmadetopicsforlengthyandspiriteddebatebytheGreeks. .BASICELEMENTSOFGEOMETRY32FUNDAMENTALSOFCOLLEGEGEOMETRYinaroom,pupilsenrolledinageometryclass,wordsintheEnglishlanguage,ThustheGreeksbecameskilledintheartoflogicandcriticalthinking.grainsofsandonabeach,etc.TheseobjectsmayalsobedistinguishableAmongthemoreprominentGreekscontributingtothisadvancementwereobjectsofourintuitionorintellect,suchaspoints,lines,numbers,andlogicalThalesofMiletUs(640-546B.C.),Pythagoras,apupilofThales(580?-500B.C.),possibilities.TheimportantfeatureofthesetconceptisthatthecollectionPlato(429-348B.C.),Archimedes(287-212B.C.),andEuclid(about300B.C.).ofobjectsistoberegardedasasingleentity.Itistobetreatedasawhole.Euclid,whowasateacherofmathematicsattheUniversityofAlexandria,Otherwordsthatconveytheconceptofsetare"group,""bunch,""class,"wrotethefirstcomprehensivetreatiseongeometry.Heentitledhistext"aggregate,""covey,"and"flock.""Elements."MostoftheprinciplesnowappearinginamoderntextwereTherearethreewaysofspecifyingaset.OneistogivearulebywhichitpresentinEuclid's"Elements."Hisworkhasservedasamodelformostofcanbedeterminedwhetherornotagivenobjectisamemberoftheset;thatthesubsequentbookswrittenongeometry.is,thesetisdescribed.Thismethodofspecifyingasetiscalledtherulemethod.Thesecondmethodistogiveacompletelistofthemembersofthe1.2.Whystudygeometry?ThestudentbeginningthestUdyofthistextset.Thisiscalledtherostermethod.Athirdmethodfrequentlyusedformaywellask,"Whatisgeometry?WhatcanIexpecttogainfromthissetsofrealnumbersistographthesetonthenumberline.ThemembersofstUdy?"asetarecalleditselements.Thus"members"and"elements"canbeusedManyleadinginstitutionsofhigherlearninghaverecognizedthatpositiveinterchangeably.benefitscanbegainedbyallwhostudythisbranchofmathematics.ThisisItiscustomarytousebraces{}tosurroundtheelementsofaset.Forevidentfromthefactthattheyrequirestudyofgeometryasaprerequisitetoexample,{I,3,5,7}meansthesetwhosemembersaretheoddnumbers1,3,matriculationinthoseschools.5,and7.{Tom,Dick,Harry,Bill}mightrepresentthememhprsofavocalisanessentialpartofthetrainingofthesuccessfulquartet.Acapitalletterisoftenusedtonameorrefertoaset.Thus,weengineer,scientist,architect,anddraftsman.Thecarpenter,machmlst~couldwriteA={I,3,5,7}andB={Tom,Dick,Harry,Bill}.tinsmith,stonecutter,artist,anddesignerallapplythefactsofgeometryinAsetmaycontainafinitenumberofelements,oraninfinitenumberoftheirtrades.Inthiscoursethestudentwilllearnagreatdealaboutgeometricelements.Afinitesetwhichcontainsnomembersistheemptyornullset.figuressuchaslines,angles,triangles,circles,anddesignsandpatternsofThesymbolforanullsetise'or{}.Thus,{evennumbersendingin5}manykinds.=f1.Asetwithadefinitenumber*ofmembersisa.finiteset.Thus,{5}isaOneofthemostimportantobjectivesderivedfromastUdyofgeometryfillitesetofwhich5istheonlyelement.Whenthesetcontainsmanyele-ismakingthestudentbemorecriticalinhislistening,reading,andthinking.ments,itiscustomarytoplaceinsidethebracesadescriptionofthemembersInstudyinggeometryheisledawayfromthepracticeofblindacceptanceofoftheset,e.g.{citizensoftheUnitedStates}.Asetwithaninfinitenumberstatementsandideasandistaughttothinkclearlyandcriticallybeforeform-ofelementsistermedaninfiniteset.ThenaturalnumbersI,2,3,....formingconclusions.aninfiniteset.{a,2,4,6,...}meansthesetofallnonnegativeevennumbers.TherearemanyotherlessdirectbenefitsthestudentofgeometrymayIt,too,isaninfiniteset.gain.AmongtheseonemustincludetrainingintheexactuseoftheEnglishInmathematicsweusethreedots(...)intwodifferentwaysinlistingthelanguageandintheabilitytoanalyzeanewsitUationorproblemintoitsbasicelementsofaset.Forexampleparts,andutilizingperseverence,originality,andlogicalreasoninginsolvingtheproblem.AnappreciationfortheorderlinessandbeautyofgeometricRuleformsthataboundinman'sworksandofthecreationsofnaturewillbeaRosterby-productofthestudyofgeometry.Thestudentshouldalsodevelopan1.{integersgreaterthan10andlessthan1O0}{ll,12,13,...,99}Herethedots...mean"andsoonuptoawarenessofthecontributionsofmathematicsandmathematicianstoourandincluding."cultureandcivilization.2.{integersgreaterthan10}{ll,12,13,...}1.3.Setsandsymbols.Theideaof"set"isofgreatimportanceinmathe-Herethedots...mean"andsoonindefinitely."matics.Allofmathematicscanbedevelopedbystartingwithsets.Theword"set"isusedtoconveytheideaofacollectionofobjects,usually*Zeroisadefinitenumber.withsomecommoncharacteristic.Theseobjectsmaybepiecesoffurniture------------ .4BASICELEMENTSOFGEOMETRY5FUNDAMENTALSOFCOLLEGEGEOMETRYTosymbolizethenotionthat5isanelementofsetA,weshallwrite5EA.21.{a,e,i,o,u}22.{a,b,c,...,z}If6isnotamemberofsetA,wewrite6~A,read"6isnotanelementofset23.{red,orange,yellow,green,blue,violet}24.{}A.25.{2,4,6,8,1O}26.{3,4,5,...,50}"27.{-2,-4,-6,...}28.{-6,-4,-2,0,2,4,6}Exercises1.4.Relationshipsbetweensets.TwosetsareequalifandonlyiftheyhaveInexercises1-12itisgiven:thesameelements.TheequalitybetweensetsAandBiswrittenA=B.TheinequalityoftwosetsiswrittenA¥'B.Forexample,letsetAbeA={1,2,3,4,5}.B={6,7,8,9,10}.{wholenumbersbetweenItand6t}andletsetBbe{wholenumbersbetweenC={I,2,3,...,1O}.D={2,4,6,...}.Hand6t}.ThenA=Bbecausetheelementsofbothsetsarethesame:E=f).F={O}.2,3,4,5,and6.Here,then,isanexampleoftwoequalsetsbeingdescribedG={5,3,2,1,4}.H={I,2,3,...}.intwodifferentways.Wecouldwrite{daysoftheweek}or{Sunday,Mon-day,Tuesday,Wednesday,Thursday,Friday,Saturday}astwowaysof1.HowmanyelementsareinC?inE?describingequalsets.2.GivearuledescribingH.Oftenseveralsetsarepartsofalargerset.Thesetfromwhichallother3.DoEandFcontainthesameelements?setsaredrawninagivendiscussioniscalledtheuniversalset.Theuniversal4.DoAandGcontainthesameelements?set,whichmaychangefromdiscussiontodiscussion,isoftendenotedbythe5.WhatelementsarecommontosetAandsetC?letterU.Intalkingaboutthesetofgirlsinagivengeometryclass,the6.WhatelementsarecommontosetBandsetD?universalsetUmightbeallthestudentsintheclass,oritcouldbeallthe7.Whichofthesetsarefinite?membersofthestudentbodyofthegivenschool,orallstudentsinallschools,8.Whichofthesetsareinfinite?andsoon.9.WhatelementsarecommontoAandB?Schematicrepresentationstohelpillustratepropertiesofandoperations10.WhatelementsareeitherinAorCorinboth?withsetscanbeformedbydrawingVenndiagrams(seeFigs.l.laandl.lb).11.InsertinthefollowingblankspacesthecorrectsymbolEor~.Here,pointswithinarectanglerepresenttheelementsoftheuniversalset.(a)3~A(b)3~D(c)O~FSetswithintheuniversalsetarerepresentedbypointsinsidecirclesencloser!(d)U~E(e)~~H(f)1O()2~Dbytherectangle.12.GivearuledescribingF.Weshallfrequentlybeinterestedinrelationshipsbetweentwoormore13-20.Usetherostermethodtodescribeeachofthefollowingsets.sets.ConsiderthesetsAandBwhereExample.{wholenumbersgreaterthan3andlessthan9}Solution.{4,5,6,7,8}A={2,4,6}andB={I,2,3,4,5,6}.13.{daysoftheweekwhosenamesbeginwiththeletterT}14.{evennumbersbetween29and39}15.{wholenumbersthatareneithernegativeorpositive}UU16.{positivewholenumbers}AllbooksAllpeople17.{integersgreaterthan9}18.{integerslessthanI}19.{monthsoftheyearbeginningwiththeletterJ}20.{positiveintegersdivisibleby3}821-28.Usetherulemethodtodescribeeachofthefollowingsets.Example.{California,Colorado,Connecticut}(a)(b)Solution.{memberstatesoftheUnitedStateswhosenamesbeginwiththeletterC}Fig.I.I.-.-----------------....-..-..---- .BASICELEMENTSOFGEOMETRY76FUNDAMENTALSOFCOLLEGEGEOMETRYWhentwosetshavenoelementsincommontheyaresaidtobedisjointDefinition:ThesetAisasubsetofsetBif,andonlyif,everyelementofsetsormutuallyexclusivesets.setAisanelementofsetB.Thus,intheaboveillustrationAisasubsetofB.(c)IfF={a,1,2,3,...}andG={0,-2,-4,-6,...},thenFnG={O}.WewritethisrelationshipACBorB:::)A.Intheillustrationtherearemore(d)GivenAisthesetofallbachelorsandBisthesetofallmales.ThenelementsinBthaninA.ThiscanbeshownbytheVenndiagramofFig.1.2.Notice,however,thatourdefinitionofsubsetdoesnotstipulateitmustcon-AnB=A.HereAisasubsetofB.tainfewerelementsthandoesthegivenset.ThesubsetcanhaveexactlytheCareshouldbetakentodistinguishbetweenthesetwhosesolememberissameelementsasthegivenset.Insuchacase,thetwosetsareequalandthenumberzeroandthenullset(seebandcabove).Theyhavequitedistincteachisasubsetoftheother.Thus,anysetisasubsetofitself.anddifferentmeanings.Thus{a}=pD.Thenullsetisemptyofanyele-Illustrationsments.Zeroisanumberandcanbeamemberofaset.Thenullsetisa(a)GivenA={I,2,3}andB={I,2}.ThenBCA.subsetofallsets.(b)GivenR={integers}andS={oddintegers}.ThenSCR.TheintersectionoftwosetscanbeillustratedbyaVenndiagram.The(c)GivenC={positiveintegers}andD={I,2,3,4,...}.ThenCCDshadedareaofFig.1.4representsAnB.andDCC,andC=D.vL'vFig.1.4.AnB.Fig.1.5.AUB.Fig.1.2.ACB.Fig.1.3.Definition:TheunionoftwosetsPandQisthesetofallelementsthatWhenAisasubsetofauniversalsetU,itisnaturaltothinkofthesetcom-belongtoeitherPorQorthatbelongtobothPandQ.posedofallelementsofUthatarenotinA.Thissetiscalledthecomple-TheunionofsetsPandQissymbolizedbyPUQandisread"PunionQ"mentofAandisdenotedbyA'.Thus,ifUrepresentsthesetofintegersandor"PcupQ."TheshadedareaofFig.1.5representstheVenndiagramofAthesetofnegativeintegers,thenA'isthesetofnonnegativeintegers,AUB.i.e.,A'={O,1,2,3,...}.TheshadedareaofFig.1.3illustratesA'.Illustrations:(a)IfA={I,2,3}andB={l,3,5,7},thenAUB={l,2,3,5,7}.1.5.Operationsonsets.WeshallnextdiscusstwomethodsforgeneratingNote.Individualelementsoftheunionarelistedonlyonce.newsetsfromgivensets.(b)IfA={wholeevennumbersbetween2tand5}andB={wholenumbersDefinition:TheintersectionoftwosetsPandQisthesetofallelementsbetween3tand6t},thenAUB={4,5,6}andAnB={4}.(c)IfP={allbachelors}andQ={allmen},thenPUQthatbelongtobothPandQ.=Q.TheintersectionofsetsPandQissymbolizedbyPnQandisread"PExample.DrawaVenndiagramtoillustrate(R'n5')'inthefigure.intersectionQ"or"PcapQ."Solution(a)ShadeR'.Illustrations:(b)Addashadefor5'.(a)IfA={1,2,3,4,5}andB={2,4,6,S,1O},thenAnB={2,4}.R'n5'isrepresentedbytheregioncommontotheareaslasheduptothe(b)IfD={I,3,5,...}andE={2,4,6,...},thenDEn=fj.--------------------------------------- BASICELEMENTSOFGEOMETRY98FUNDAMENTALSOFCOLLEGEGEOMETRYu7.InthefollowingstatementsPandQrepresentsets.Indicatewhichofthefollowingstatementsaretrueandwhichonesarefalse.(a)PnQisalwayscontainedinP.(b)PUQisalwayscontainedinQ.(c)PisalwayscontainedinPUQ.(])(d)QisalwayscontainedinPUQ.(e)PUQisalwayscontainedinP.(f)PnQisalwayscontainedinQ.(g)PisalwayscontainedinPnQ.(a)R'(h)QisalwayscontainedinPnQ.u(i)IfP=>Q,thenPnQ=P.(j)IfP=>Q,thenPnQ=Q.(k)IfPCQ,thenPUQ=P.(1)IfPCQ,thenPUQ=Q.rn[i]8.Whatisthesolutionsetforthestatementa+2=2,i.e.,thesetofallsolutions,ofstatementa+2=2?9.Whatisthesolutionsetforthestatementa+2=a+4?10.LetDbethesetoforderedpairs(x,y)forwhichx+y=5,andletEbethesetoforderedpairs(x,y)forwhichx-y=1.WhatisDnE?(b)(e)(R'nS')'R'nS'11-30.Copyfiguresanduseshadingtoillustratethefollowingsets.rightandtheareaslasheddowntotheright.(R'nS')'isalltheareainII.RUS.12.RnS.UthatisnotinR'nS'.13.(RnS)'.14.(RUS)'.u(c)Thesolutionisshadedinthelastfigure.15.R'.16.S'.Wenotethat(R'nS')'=RUS.17.(R')'.18.R'US'.b-19.R'nS'.20.(R'nS')'.Exercises21.RUS.22.RnS.2~LR'nS'.24.R'US'.1.LetA={2,3,5,6,7,9}andB={3,4,6,8,9,1O}.25.RUS.26.RnS.(a)WhatisAnB?(b)WhatisAUB?27.R'nS'.28.R'US'.2.LetR={I,3,5,7,...}andS={O,2,4,6,...}.2~).R'US.30.RUS'.Exs.1l-20.(a)WhatisRnS?(b)WhatisRUS?3.LetP={I,2,3,4,...}andQ={3,6,9,I2,...}.(a)WhatisPnQ?(b)WhatisPUQ?uu4.({I,3,5,7,9}n{2,3,4,5})U{2,4,6,8}=?5.Simplify:{4,7,8,9}U({I,2,3,...}n{2,4,6,...}).6.Considerthefollowingsets.A={studentsinyourgeometryclass}.B={malestudentsinyourgeometryclass}.00C={femalestudentsinyourgeometryclass}.D={membersofstudentbodyofyourschool}.Whatare(a)AnB;(b)AUB;(c)Bnc;(j)AUD?Ex.I.21-24.Exs.25-30.------------------------------------ BASICELEMENTSOFGEOMETRY1110FUNDAMENTALSOFCOLLEGEGEOMETRYis90.Theexpression"ifandonlyif"willbeusedsofrequentlyinthistext1.6.Needfordefinitions.Instudyinggeometrywelearntoprovestate-thatwewillusetheabbreviation"iff"tostandfortheentirephrase.mentsbyaprocessofdeductivereasoning.Welearntoanalyzeaproblemintermsofwhatdataaregiven,whatlawsandprinciplesmaybeacceptedas1.7.Needforundefinedterms.Therearemanywordsinusetodaythataretrueand,bycareful,logical,andaccuratethinking,welearntoselectasolu-difficulttodefine.Theycanonlybedefinedintermsofotherequallyun-tiontotheproblem.Butbeforeastatementingeometrycanbeproved,wedefinableconcepts.Forexample,a"straightline"isoftendefinedasalinemustagreeoncertaindefinitionsandpropertiesofgeometricfigures.It"nopartofwhichiscurved."Thisdefinitionwillbecomeclearifwecanisnecessarythatthetermsweuseingeometricproofshaveexactlythesamedefinethewordcurved.However,ifthewordcurvedisthendefinedasameaningtoeachofus.line"nopartofwhichisstraight,"wehavenotrueunderstandingoftheMO,stofusdonotreflectonthemeaningsofwordswehearorreadduringdefinitionoftheword"straight."Suchdefinitionsarecalled"circularthecourseofaday.Yet,often,amorecriticalreflectionmightcauseustodefinitions."Ifwedefineastraightlineasoneextendingwithoutchangeinwonderwhatreallywehaveheardorread.direction,theword"direction"mustbeunderstood.Indefiningmathe-Acommoncauseformisunderstandingandargument,notonlyingeometrymaticalterms,westartwithundefinedtermsandemployasfewaspossibleofbutinallwalksoflife,isthefactthatthesamewordmayhavedifferentthosetermsthatareindailyuseandhaveacommonmeaningtothereader.meaningstodifferentpeople.Inusinganundefinedterm,itisassumedthatthewordissoelementaryWhatcharacteristicsdoesagooddefinitionhave?Whencanwebecertainthatitsmeaningisknowntoall.Sincetherearenoeasierwordstodefinethedefinitionisagoodone?Noonepersoncanestablishthathisdefinitiontheterm,noeffortismadetodefineit.Thedictionarymustoftenresorttoforagivenwordisacorrectone.Whatisimportantisthatthepeople"defining"awordbyeitherlistingotherwords,calledsynonyms,whichhaveparticipatinginagivendiscussionagreeonthemeaningsofthewordinthesame(oralmostthesame)meaningasthewordbeingdefinedorbyquestionand,oncetheyhavereachedanunderstanding,nooneofthegroupdescribingtheword.maychangethedefinitionofthewordwithoutnotifyingtheothers.Wewillusethreeundefinedgeometrictermsinthisbook.Theyare:Thiswillespeciallybetrueinthiscourse.Onceweagreeonadefinitionpoint,straightline,andplane.Wewillresorttosynonymsanddescriptionsstatedinthistext,wecannotchangeittosuitourselves.Ontheotherhand,ofthesewordsinhelpingthestudenttounderstandthem.thereisnothingsacredaboutthedefinitionsthatwillfollow.Theymight1.8.Pointsandlines.Beforewecandiscussthevariousgeometricfigureswellbeimprovedon,aslongaseveryonewhousestheminthistextagreesto,[:,setsofpoints,wewillneedtoconsiderthenatureofapoint.""Vhatisait.point?Everyonehassomeunderstandingoftheterm.AlthoughwecanAgooddefinitioningeometryhastwoimportantproperties:representapointbymarkingasmalldotonasheetofpaperoronablack-board,itcertainlyisnotapoint.Ifitwerepossibletosubdividethemarker,I.Thewordsinthedefinitionmustbesimplerthanthewordbeingde-thensubdivideagainthesmallerdots,andsoonindefinitely,westillwouldfinedandmustbeclearlyunderstood.nothaveapoint.Wewould,however,approachaconditionwhichmostof2.Thedefinitionmustbeareversiblestatement.usassigntothatofapoint.Euclidattemptedtodothisbydefiningapointasthatwhichhaspositionbutnodimension.However,thewords"position"Thus,forexample,if"rightangle"isdefinedas"ananglewhosemeasureand"dimension"arealsobasicconceptsandcanonlybedescribedbyusingis90,"itisassumedthatthemeaningofeachterminthedefinitionisclearandcirculardefinitions.that:Wenameapointbyacapitalletterprintedbesideit,aspoint"A"inFig.1.6.OthergeometricfigurescanbedefinedintermsofsetsofpointswhichsatisfyI.Ifwehavearightangle,wehaveananglewhosemeasureis90.certainrestrictingconditions.2.Conversely,ifwehaveananglewhosemeasureis90,thenwehavearight,Weareallfamiliarwithlines,butnoonehasseenone.Justaswecan~~.Irepresentapointbyamarkerordot,wecanrepresentalinebymovingthetipofasharpenedpencilacrossapieceofpaper.ThiswillproduceanThus,theconverseofagooddefinitionisalwaystrue,althoughtheconverse:approximationforthemeaninggiventotheword"line."Euclidattemptedofotherstatementsarenotnecessarilytrue.Theabovestatementandits:todefinealineasthatwhichhasonlyonedimension.Here,again,heusedconversecanbewritten,"Anangleisarightangleif,andonlyif,itsmeasure BASICELEMENTSOFGEOMETRY1312FUNDAMENTALSOFCOLLEGEGEOMETRY13IIInA13i~~IIIIIFig.1.7.I§Fig.1.6.//.J'---/theundefinedword"dimension"inhisdefinition.AlthoughwecannotCubeSpheredefinetheword"line,"werecognizeitasasetofpoints.Onpage11,wediscusseda"straightline"asonenopartofwhichis"curved,"orasonewhichextendswithoutchangeindirections.Thefailuresoftheseattemptsshouldbeevident.However,theword"straight"isanabstractionthatisgenerallyusedandcommonlyunderstoodasaresultofmanyobserva-tionsofphysicalobjects.Thelineisnamedbylabelingtwopointsonitwithcapitallettersorbyonelowercaseletternearit.ThestraightlineinFig.1.7isread"lineAB"or"linel."LineABisoftenwritten"AE."Inthisbook,unlessotherwisestated,whenweusetheterm"line,"wewillhaveinmindtheconceptofastraightline.----------IfBEl,AEI,andA=1=B,wesaythatlisthelinewhichcontainsAandB.Twopointsdeterminealine(seeFig.'1.7).ThusAB=BA.CylinderConePyramidTwostraightlines~intersectinonlyonepoint.InFig.1.6,ABnXC={A}.WhatisABnBC?Fig.1.9.IfwemarkthreepointsR,S,andT(Fig.1.8)allonthesameline,weseethatRS=IT.Threeormorepointsarecollinearifftheybelongtotherepresentsplane1'.11'1orplaneM.Wecanthinkoftheplaneasbeingmadesameline.upofaninfinitenumberofpointstoformasurfacepossessingnothicknessbuthavinginfinitelengthandwidth.Twolineslyinginthesameplanewhoseintersectionisthenullsetaresaidstobeparallellines.Iflinelisparalleltolinem,thenlnm=(}.InFig.1.10,,llJisparalleltoDCandADisparalleltoBe.ThedrawingsofFig.1.12andFig.1.13illustratevariouscombinationsofpoints,lines,andplanes.Fig.1.8.E1.9.Solidsandplanes.CommonexamplesofsolidsareshowninFig.1.9.ThegeometricsolidshowninFig.1.10hassixfaceswhicharesmoothandcflat.Thesefacesaresubsetsofplanesurfacesorsimplyplanes.Thesurfaceofablackboardorofatabletopisanexampleofaplanesurface.Aplanecanbethoughtofasasetofpoints..r'-------/'Definition.Asetofpoints,allofwhichlieinthesameplane,aresaidAntobecoplanar.PointsD,C,andEofFig.1.10arecoplanar.AplanecanbeFig.1.10.Fig.l.ll.namedbyusingtwopointsorasinglepointintheplane.Thus,Fig.1.11 14FUNDAMENTALSOFCOLLEGEGEOMETRY,BASIC5.CanalinealwaysELEMENTSOFGEOMETRY15be6.CanaplanealwaysPassedthroughanythreedistinctpoints?b~passed7.CantwoplaneseverIl1tersectthroughanythreedistinctpoints?8.CanthreeplanesintersectinthesalTleinasinglepoint?straightline?9-17.Refertothefigureandindicatewhichofthefollowingtrueandwhicharefalse.statementsare9.PlaneABintersectsplaneCDinlinel.10.PlaneABpassesthroughlinel.I].PlaneABpassesthroughEF.~Fig.I.I2.12.PlaneCDpassesthroughY.LinerintersectsplaneR.]3.PEplaneCD.PlaneRcontainslinelandm.14.(planeAB)n(planeCD)=EF.--PlaneRpassesthroughlineslandm.15.lnEF=G.PlaneRdoesnotpassthroughliner.]6.(planeCD)n1=G.PlaneMNandPlaneRSintersectinAB.]7.(planeAB)nEF~=EF.~PlaneMNandPlaneRSbothpassthroughAlJ.18-38.Drawpictures(ifpossible)thatillustratethesituationsdescribed.ABliesinbothplanes.18.landmaretwolinesandlnm={P}.ABiscontainedinplanesMNandRS.19.1andmaretwolines,PEl,REl,SEmandRS~AB,~~20.CandA¥'PRoExercises¥'B.21.RESf.1.Howmanypointsdoesalinecontain?2.Howmanylinescanpassthroughagivenpoint?3.Howmanylinescanbepassedthroughtwodistinctpoints?4.Howmanyplanescanbepassedthroughtwodistinctpoints?,8iRBAFig.I.I3.Ex..'.9-17. BASICELEMENTSOFGEOMETRY1716FUNDAMENTALSOFCOLLEGEGEOMETRY(}3115467711622.randsaretwolines,andI'ns=(}.(,I-rI-~-LL1~~ilL23.randsaretwolines,andI'ns¥-.-4-32-101234(}.24.P~fl,PEl,and1nKl=Fig.1.15.25.R,S,andTarethreepointsandTE(RTns'f).26.randsaretwolines,A¥-B,and{A,B}C(I'ns).Wehavenowexpandedthepointsonthelinetorepresentallrealrational27.P,Q,R,andSarefourpoints,QEPH,andRE([S.++numbers.28.P,Q,R,andSarefournoncollinearpoints,QEPItandQEPS.29.A,B,andCarethreenoncollinearpoints,A,B,andDarethreecollinearDefinition:Arationalnumberisonethatcanbeexpressedasaquotientofpoints,andA,C,andDarethreecollinearpoints.integers.30.I,m,andnarethreelines,andPE(mnn)nI.Itcanbeshownthateveryquotientoftwointegerscanbeexpressedasa31.I,m,andnarethreelines,A¥-B,and{A,B}C(lnm)nn.repeatingdecimalordecimalthatterminates,andeverysuchdecimalcanbe(lnm)U(nnm).32.I,m,andnarethreelines,A¥-B,and{A,B}=writtenasanequivalentindicatedquotientoftwointegers.Forexample,33.A,B,andr:arethreecollinearpoints,C,D,andEarethreenoncollinear13/27=0.481481...and1.571428571428...=1117arerationalnumbers.points,andEEAB.Therationalnumbersformaverylargeset,forbetweenanytworational34.(planeRS)n(planeMN)=AB.numbersthereisathirdone.Therefore,thereareaninfinitenumberof(}35.(planeAB)n(planeCD)=.pointsrepresentingrationalnumbersonanygivenscaledline.However,{P}.36.line1CplaneAB.linemCplaneCD.lnm=(}therationalnumbersstilldonotcompletelyfillthescaledline.37.(planeAB)n(planeCD)=I.linemEplaneCD.1nm=.(}.Definition:Anirrationalnumberisonethatcannotbeexpressedasthe38.(planeAB)n(planeCD)=t.linemEplaneCD.lnrn¥-quotientoftwointegers(orasarepeatingorterminatingdecimal).1.10.Realnumbersandthenumberline.ThefirstnumbersachildExamplesofirrationalnumbersareV2,-y'3,Y5,and1T.Approximatelearnsarethecountingornaturalnumbers,e.g.,{I,2,3,...}.Thenaturallocationsofsomerationalandirrationalnumbersonascaledlineareshowninnumbersareinfinite;thatis,givenanynumber,howeverlarge,thereisalwaysFig.1.16.anothernumberlarger(add1tothegivennumber).ThesenumberscanbeTheunionofthesetsofrationalandirrationalnumbersformthesetofrepresentedbypointsonaline.Placeapoint0onthelineX'X(Fig.1.14).realnumbers.ThelinethatrepresentsalltherealnumbersiscalledtherealThepoint0willdividethelineintotwoparts.Next,letAbeapointonX'Xnumberline.ThenumberthatispairedwithapointonthenumberlineistotherightofO.Then,totherightofA,markoffequallyspacedpointsB,calledthecoordinateofthatpoint.C,D,....ForeverypositivewholenumbertherewillbeexactlyonepointWesummarizebystatingthattherealnumberlineismadeupofaninfinitetotherightofpointO.Conversely,eachofthesepointswillrepresentonlysetofpointsthathavethefollowingcharacteristics.onepositivewholenumber.Inlikemanner,pointsR,S,T,...canbemarkedofftotheleftofpoint0toI.Everypointonthelineispairedwithexactlyonerealnumber.representnegativewholenumbers.2.Everyrealnumbercanbepairedwithexactlyonepointontheline.ThedistancebetweenpointsrepresentingconsecutiveintegerscanbeWhen,giventwosets,itispossibletopaireachelementofeachsetwithdividedintohalves,thirds,fourths,andsoon,indefinitely.Repeatedexactlyoneelementoftheother,thetwosetsaresaidtohaveaone-to-onedivisionwouldmakeitpossibletorepresentallpositiveandnegativefractionscorrespondence.Wehavejustshownthatthereisaone-to-onecorrespon-withpointsontheline.NoteFig.1.15forafewofthenumbersthatmightbederKebetweenthesetofrealnumbersandthesetofpointsonaline.assignedtopointsontheline.4=60-3-,(3-l1-vi-{/57r3"2wVUTSR0ABCDEFXX'IIIIIIIIIIIII:.---5a-4-3-2-101b23c456-543-2-1023456Fig.1.17.Fig.1.19.1.11.Orderandthenumberline.Allofusatonetimeoranotherengageinthesmallernumberrepresentedbythesetwopointsfromthelarger.Thus,comparingsizesofrealnumbers.SymbolsareoftenusedtoindicatetheinFig.1.19:Thedistance/romTtoV=5-(-I)=6.relativesizesofrealnumbers.Considerthefollowing.ThedistancefromStoT=(-1)-(-3)=2.SymbolMeaningThedistancefromQtoR=3-(-5)=8.a=baequalsbAnotherwaywecouldstatetheaboverulecouldbe:"Subtracttheco-a¥=baisnotequaltobordinateoftheleftpointfromthatofthepointtotheright."However,thisa>baisgreaterthanbrulewouldbedifficulttoapplyifthecoordinateswereexpressedbyplaceabandba,thepointrepresentingthenumberbwillbelocatedtotheConsiderthefollowingillustrationsofthepreviousexamples.rightofthepointonthenumberlinerepresentingthenumbera(seeFig.1.17).Conversely,ifpointSistotherightofpointR,thenthenumberwhichColumn1Column2isassignedtoSmustbelargerthanthatassignedtoR.Inthefigure,ba.15-(-1)1=101=01(-1)-(+5)j=I-oj=0Whenwewriteorstatea=bwemeansimplythata.andbaredifferent1(-1)-(-3)1=121=2i(-3)-(-1)namesforthesamenumber.Thus,pointswhichrepresentthesamenumber1=1-21=213-(-5)1=181=81(-5)-(+3)1=1-81=8onanumberlinemustbeidentical.1.12.Distancebetweenpoints.Ofteninthestudyofgeometry,wewillbeThus,wenotethattofindthedistancebetweentwopointsweneedonlytoconcernedwiththe"distancebetweentwopoints."ConsiderthenumbersubtractthecoordinatesineitherorderandthentaketheabsolutevalueoflineofFig.1.18wherepointsA,P,B,C,respectivelyrepresenttheintegersthedifference.Ifaandbarethecoordinatesoftwopoints,thedistancebetweenthe-3,0,3,6.WenotethatAandBarethesamedistancefromP,namely3.pointscanbeexpressedeitherbyla-blorIb-al.NextconsiderthedistancebetweenBandC.Whilethecoordinatesdifferintheseandtheprevioustwocases,itisevidentthatthedistancebetweentheExercisespointsisrepresentedbythenumber3.1.WhatisthecoordinateofB?ofD?Howcanwearriveatarulefordeterminingdistancebetweentwopoints?2.WhatpointlieshalfwaybetweenBandD?Wecouldfindthedi~tancebetweentwopointsonascaledlinebysubtracting3.Whatisthecoordinateofthepoint7unitstotheleftofD?APBCABCPDEFABo--~-----(e)EX5.1-12.Fig.1.24.(a)LineAB.(b)Half-lineAB.Half-lineBA.6.AreCAandcDoppositerays?7.IsCEliD?8.WhatisCAnliD?Definition:IfAandBarepointsoflinel,thenthesetofpointsconsisting9.WhatisliAnBD?ofAandallthepointswhichareonthesamesideofAasisBistherayfromA10.WhatisABUBG?throughB.ThepointAiscalledtheendPointofrayAB.II.WhatisARUif(?ThesymbolfortherayfromAthroughBisAB(Fig.1.25a)andisread12.Whatist:BnJD?"rayAB."ThesymbolfortherayfromBthroughA(Fig.1.25b)isBA.13-32.Drawpictures(ifpossible)thatillustratethesituationsdescribedinthefollowingexercises.AB...;.13.BisbetweenAandC,andCisbetweenAandD.---------..14.A,B,C,andDare(a)fourcollinearpoints,AisbetweenCandD,andDisbetweenAandB.15.H.EITandR~IT.lb.1'7Qe[[S.~0-0-ABII.QPeRG......--------18.BEA'(;andCisbetweenBandD.(b)19.PQ=PRuJiQ20.TERSandSElIT.Fig.1.25.(a)RayAB.(b)Ray.BA.2.PQ=PRuJ5Q.22.AJjnCD={E}.Definition:BAandBCarecalledoppositeraysiffA,B,andCarecollinear<)(Q--:)o>~0---+~~~3.Pg,PH.,andPSarethreehalf-lines,andQRnPS¥£).pointsandBisbetweenAandC(Fig.1.26).24.PQ,PR,andPSarethreehalf-lines,andQRn/is=£).ItwillbeseenthatpointsAandBofFig.1.26determineninegeometric+--+~n7i')'~:J.PQ=rRUPI.!.~Ires:AiJ,AB,AB,~AB,B11fA~herat2~pposite/f(3,andt~erayo'pposite26.PQ=~u([R.BA.TheunionofBAandBCisBC(orAC).ThemtersectlOnofBAand~~27.PQ=PQu@.ABisAB.28.P,Q,andRarethreecollinearpoints,PE([fl.,andR~P"Q.29.l,m,andnarethreedistinctlines,lnm=,0,mnn=,0.30.I,m,andnarethreedistinctlines,lnm=,0,mnn=YI,lnn¥,0.31.REK1andLE1fH.32.DEJKandFEi5ltFig.1.26. BASICELEMENTSOFGEOMETRY2524FUNDAMENTALSOFCOLLEGEGEOMETRY1.14.Angles.ThefiguredrawninFig.1.27isarepresentationofanangle.uDefinitions:Anangleistheunionoftworayswhichhavethesameendpoint.Theraysarecalledthesidesoftheangle,andtheircommonend-H'~pointiscalledthevertexoftheangle.~Hz8AFig.1.29.Fig.1.30.Fig.1.27.half-planesHIandHz(Fig.1.29).ThetwosetsofpointsHIandHzareThesymbolforangleisL;theplural,,6,.Therearethreecommonwayscalledsides(orhalf-Planes)oflineI.Thelineliscalledtheedgeofeachhalf-plane.Noticethatahalf-planedoesnotcontainpointsofitsedge;thatofnaminganangle:(1)bythreecapitalletters,themiddleletterbeingthe~is,ldoesnotlieineitherofthetwohalf-planes.Wecanwritethisfactvertexandtheothertwobeingpointsonthesidesoftheangle,asLABC;.(2)byasinglecapitalletteratthevertexifitisclearwhichangleismeant,asIIasHIn1=fjandHznl=fj.Ahalf-planetogetherwithitsedgeiscalledLB;and(3)byasmallletterintheinterioroftheangle.InadvancedworkinaclosedhallPlane.TheplaneU=HIUIUHz.mathematics,thesmallletterusedtonameanangleisusuallyaGreekletter,IIftwopointsPandQofplaneUlieinthesamehalf-plane,theyaresaidasL.ThestudentwillfindthelettersoftheGreekalphabetintheappendixtolieonthesamesideofthelineIwhichdividestheplaneintothehalf-planes.IInthiscasePQnl=~.IfPliesinonehalf-planeofUandRintheotherofthisbook.Thestudentshouldnotethatthesidesofanangleareinfinitelylongintwo'=.~(Fig.1.30),theylieonopposite.idesof!.HerePRnI#~.directions.Thisisbecausethesidesofananglearerays,notsegments.InFig.1.28,LAOD,LBOE,andLCOFallrefertothesameangle,LO.1.16.Interiorandexteriorofanangle.ConsiderLABC(Fig.1.31)lyinginplaneU.LineABseparatestheplaneintotwohalf-planes,oneofwhich1.15.Separationofaplane.Apointseparatesalineintotwohalf-lines.containsC.LineBCalsoseparatestheplaneintotwohalf-planes,oneofInasimilarmanner,wecanthinkofalineseparatingaplaneUintotwowhichcontainsA.Theintersectionofthesetwohalf-planesistheinterioroftheLABC.Definitions:ConsideranLABClyinginplaneU.TheinterioroftheangleisthesetofallpointsoftheplaneonthesamesideofAJjasCandonthesamesideof1fCasA.TheexteriorofLABCisthesetofallpointsofUthatdonotlieontheinterioroftheangleorontheangleitself.AcheckofthedefinitionswillshowthatinFig.1.31,pointPisintheinteriorofLABC;pointsQ,R,andSareintheexterioroftheangle.1.17.Measuresofangles.Wewillnowneedtoexpressthe"size"ofanangleinsomeway.Anglesareusuallymeasuredintermsofthedegreeunit.Fig.1.28. BASICELEMENTSOFGEOMETRY2726FUNDAMENTALSOFCOLLEGEGEOMETRYJustasarulerisusedtoestimatethemeasuresofsegments,themeasureof---ananglecanbefoundroughlywiththeaidofaprotractor.(Fig.1.33).--Fig.1.31.Fig.1.33.Aprotractor.Definition:ToeachangletherecorrespondsexactlyonerealnumberrThus,inFig.1.34,weindicatetheanglemeasuresas:between0and180.Thenumberriscalledthemeasureordegreemeasureoftheangle.mLAOB=20mLCOD=186-501or150-861=36Whilewewilldiscusscircles,radii,andarcsatlengthinChapter7,itismLAOD=86mLDOF=1150-861or186-1501=64assumedthatthestudenthasatleastanintuitiveunderstandingoftheterms.mLAOF=ISOmLBOE=Il10~201or120-1101=90Thus,tohelpthestudentbettertocomprehendthemeaningofthetermwewillstatethatifacircleisdividedinto360equalarcsandradiiaredrawnloanytwoconsecutivepointsofdivision,theangleformedatthecenterbythese1radiihasameasureofonedegree.Itisaone-degreeangle.Thesymbol1)/fordegreeis°.Thedegreeisquitesmall.Wegainaroughideaofthe"size"ofaone-degreeanglewhenwerealizethat,ifinFig.1.32(notdrawntoscale),BAandBCareeach57incheslongandACisoneinchlong,thenLABChasameasureofapproximatelyone.WecandescribethemeasureofangleABCthreeways:ThemeasureofLABCis1.mLABC=1.G~LABCisa(me-degreeangle.()~Fig.1.34.c=-lThereadershouldnotethatthemeasureofanangleismerelytheabsoluteBvalueofthediH"crencebetweennumberscorrespondingtothesidesoftheangle.Hence,assuch,itismerelyanumberandnomore.WeshouldnotFig.1.32. BASICELEMENTSOFGEOMETRY2928FUNDAMENTALSOFCOLLEGEGEOMETRYcexpressthemeasureofanangleas,letussay,30degrees.However,wewillalwaysindicateinadiagramthemeasureofananglebyinsertingthenumberwithadegreesignintheinterioroftheangle(seeFig.1.35).Thenumber45isthenumberofdegreesintheangle.Thenumberitselfiscalledthemeasureoftheangle.Bydefin-Aingthemeasureoftheangleasanumber,BABFwemakeitunnecessarytousethewordExs.1-1O.degreeortousethesymbolfordegreeinFig.1.35.mLABC=45.expressingthemeasureoftheangle.4.NamethetwosidesofLFBC.Inusingtheprotractor,werestrictourselvestoangleswhosemeasuresare5.WhatisLABDnLDBC?nogreaterthan180.Thiswillexcludethemeasuresofafiguresuchas6.WhatisLAMDnLBMC?LABCillustratedinFig.1.36.Whileweknowthatanglescanoccurwhose7.Namethreeangleswhosesidesarepairsofoppositerays.measuresaregreaterthan180,theywillnotariseinthistext.HenceLABC8.WhatisACnffD?insuchafigurewillrefertotheanglewiththesmallermeasure.Thestudy9.WhatisMAUMD?ofangleswhosemeasuresaregreaterthan180willbelefttothemoread-10.WhatisiWAUJill?vancedcoursesinmathematics.11-20.Draw(ifpossible)picturesthatillustratethesituationsdescribedineachofthefollowing.11.Iisaline.PQn1=J1.cl~.lisaline.PQnI~0.13.lisaline.PQnI~,0.Rln1=0.14.IiSZlti~;_=-t'Q;fiL=it:~--ffl-A-T==:ft.::~u_-r---------15.lisaline.PQnl=0.PRnl~0.16.1isaline.PQn1=,0.QRn1=,0.PRnl~0.17.lisaline.PQn1=0.QRnI~0.'fiRn1=0.18.IisalinewhichseparatesplaneUintohalf-planesHIandHz.PQ-.n1=0,PEHI,QEHz.Fig.1.36.19.Ideterminesthetwohalf-planeshiandhz.REI,SIi:I,liS'Chi,20.Ideterminesthetwohalf-planeshiandhz.REI,S~I,RSChi,,IThestudentmaywonderabouttheexistenceofananglewhosemeasureisExercises(B)O.Wewillassumethatsuchanangleexistswhenthetwosidesoftheangle,:coincide.Youwillnotethattheinteriorofsuchanangleistheemptyset,J1.21.Drawtwoangleswhoseinteriorshavenopointsincommon.22.IndicatethemeasureoftheangleExercises(A)inthreedifferentways.23.Byusingaprotractor,drawan~~D1.NametheangleformedbyiWDandiVfCinthreedifferentways.Canglewhosemeasureis55.2.NameLainfouradditionalways.LabeltheangleLKTR.Ex.22.3.GivethreeadditionalwaystonameliM. 30FUNDAMENTALSOFCOLLEGEGEOMETRYBASICELEMENTSOFGEOMETRY3I24.Findthevalueofeachofthefollowing:t~twoadjacentangles.InFig.1.37LAGEandLBGCareadjacentangles.(a)mLA]C.(d)mLD]B.(g)mLH]C+mLfJE.DBliesintheinteriorofLAOG.(b)mLCJE.(e)mLBJF.(h)mLHJB-mLfJD.(c)mLH]C.(f)mLCJD+mLG]D.(i)mLD]G-mLBJC.AFig.1.37.AdjacentLS.Thepairsofnonadjacentanglesformedwhentwolinesintersectare3;00oEtermedverticalangles.InFig.1.38LaandLa'areverticalanglesandsoareLj3andLj3'.Ex.24.25.DrawABCIsuchthatm(AB)=4inches.AtAdrawACsuchthatmLBAC=63.AtBdrawEDsuchthatmLABD=48.Labelthepoint{K}.WiththeaidofwheretheraysintersectasK.Thatis,ACnBb=aprotractorfmdmLAKB.26.Complete:(a)mLKPL+mLLPivlmL.-Fig.1.38.LaandLa'areverticalLS.(b)mLMPN+mLLPM=mL.(c)mLKPM-rnLLPM=mL.Asthemeasureofanangleincreasesfrom0to180thefollowingkindsof(d)mLKPN-mLl.lPN=rnL.anglesareformed:acuteangle,rightangle,obtuseangle,andstraightangle27.Withtheaidofaprotractor(seeFig.1.39).drawananglewhosemeasureis70.CallitLRST.LocateaKLMNDefinitions:Anangleisanacuteangleiffithasameasurelessthan90.pointPintheinteriorofLRSTAnangleisarightangleiffithasameasureof90.AnangleisanobtuseangleEx.26.iffitsmeasureismorethan90andlessthan180.Anangleisastraightanglesuchthatm(SPUS7)=25.iffitsmeasureisequalto180.WhatisrnLPSR?28.WiththeaidofaprotractordrawLABCsuchthatmL1BC=120.LocateActually,ourdefinitionforthestraightanglelacksrigor.SinceweapointPintheexteriorofLABCsuchthatBEPC.Findthevalueofdefinedanangleasthe"unionoftworayswhichhaveacommonendpoint,"m(BPUB/1).weknowthatthedefinitionshouldbeareversiblestatement.Therefore,we1.18.Kindsofangles.Twoanglesaresaidtobeadjacentanglesifftheyhavewouldhavetoconcludethateveryunionoftworayswhichhavethesamethesamevertex,acommonside,andtheothertwosidesarecontainedinendpointwouldproduceanangle.YetweknowthatBCUBAisAG.Weare,ineffect,thensayingthatastraightangleisastraightline.Thisweknowoppositeclosedhalf-planesdeterminedbythelinewhichcontainsthecom-isnottrue.Anangleisnotaline.monside.Theraysnotcommontobothanglesarecalledexteriorsidesof------------------------------------- 32FUNDAMENTALSOFCOLLEGEGEOMETRYBASICELEMENTSOFGEOMETRY33Candthesameshape."CongruentfigurescanbethoughtofasbeingCduplicatesofeacr.other.Definitions:Planeanglesarecongruentifftheyhavethesamemeasure.Segmentsarecongruentifftheyhavethesamemeasure.Thus,ifweknowLR'WLBAAthatmAB=mCD,wesaythatABandCDarecongruent,thatABiscongruentAcuteLRightLtoCDorthatCDiscongruenttoAB.Again,ifweknowthatmLABC=(a)(1))mLRST,wecansaythatLABCandLRSTarecongruentangles,LABCiscongruenttoLRST,orthatLRSTiscongruenttoLABC.Thesymbolswehaveusedthusfarinexpressingtheequalityofmeasuresbetweenlinesegmentsorbetweenanglesisrathercumbersome.Toover-180.comethis,mathematicianshaveinventedanewsymbolforcongruence.The~<~-)Isvmbolfor"iscongruentto"is==.Thus,thefollowingareequivalentBACBAstatements.*mAB=mCDAB==CDmLABC=mLRSTLABCObtuseLStraightL==LRST(c)(d)Definition:ThebisectorofanangleistheraywhoseendpointistheFig.1.39.vertexoftheangleandwhichdividestheangleintotwocongruentangles.TherayBDofFig.1.41bisects,oristheanglebisectorof,LABCiffDisintheHowever,sincetheterm"straightangle"isquitecommonlyusedtointeriorofLABCandLABD==LDBC.representsuchafigureasillustratedinFig.1.39d,wewillfollowthatpractice1.20.Perpendicularlinesandrightangles.Considerthefourfiguresinthisbook.Sometextscallthefigurealinearpair.*showninFig.1.42.TheyareexamplesofrepresentationsofrightanglesDefinition:IfA,B,andCarecollinearandAandperpendicularlines.candCarconoppositesidesofB,thenRAUBCisDefinition:TwolinesareperpendicularifftheyintersecttoformarightBcalledastraightanglewithBitsvertexandH"Aandangle.RaysandsegmentsaresaidtobeperpendiculartoeachotheriffBCthesides.thelinesofwhichtheyaresubsetsareperpendiculartoeachother.IDefinition:AdihedralangleisformedbytheIunionoftwohalf-planeswiththesameedge.IIEachhalf-planeiscalledafaceoftheangle(seeI1Fig.1.40).DihedralangleswillbestudiedinIIChapter14.IIJ1.19.Congruentangles.Congruentsegments.AcommonconceptindailylifeisthatofsizeBandcomparativesizes.WefrequentlyspeakofAtwothingshavingthesamesize.ThewordFi?;.1.41.Anglebisector.F"congruent"isusedingeometrytodefinewhatFi?;.1.40.Dihedralangle.weintuitivelyspeakofas"havingthe*:VlanytextswillalsousethesymbolAB=CDtomeanthatthemeasuresofthesegmentsareequal.Yourinstructormaypermitthissymbolism.However,inthistext,wewillnotusethis*Manytextbooks,also,willdefineanangleasareflexan?;leiffitsmeasureismorethan180butsymbolismforcongruenceofsegmentsuntilChapter8.Bythattime,surely,thestudentlessthan360.Wewillhavenooccasiontousesuchanangleinthistext.willnotconfuseageometricfIgurewiththatofitsmeasure. BASICELEMENTSOFGEOMETRY3534FUNDAMENTALSOFCOLLEGEGEOMETRYBcA[B~.JM""~~BCAFig.1.44.DistancefrompointtolineThus,inFig.1.44,themeasureofPMisthedistancefrompointPtoAiJ.InChapter9,wewillprovethattheperpendiculardistanceistheshortestdistancefromapointtoaline.1.22.Complementaryandsupplementaryangles.TwoanglesarecalledACJcomPlementaryanglesiffthesumoftheirmeasuresis90.ComplementaryanglescouldalsobedefinedastwoanglesthesumofwhosemeasuresequalsBCthemeasureofarightangle.InFig.1.45LaandL{3arecomplementaryangles.Eachisthecomplementoftheother.AngleaisthecomplementofL{3;andL{3isthecomplementofLa.Fig.1.42.Perpendicularlines.Thesymbolforperpendicularis.l.Thesymbolmayalsoberead"per-pendicularto."Arightangleofafigureisusuallydesignatedbyplacingasquarecornermarkl1.wherethetwosidesoftheanglemeet.Thefootoftheperpendiculartoalineisthepointwheretheperpendicularmeetstheline.Thus,Bisthefoof6ftneperpenctkuiarsiIT-hg:l:42-:----A-Aline,ray,orsegmentisperpendiculartoaplaneifitisperpendicular-~to~ryl~i!.:..theElanethatpassesthroughitsfoot.InFig.1.43,PQ.lAQ,Fig.1.45.ComplementaryLS.PQ.lAQ,PQ.lQB.1.21.Distancefromapointtoaline.ThedistancefromapointtoalineAnglesaresupplementaryanglesiffthesumoftheirmeasuresis180.Weisthemeasureoftheperpendicularsegmentfromthepointtotheline.couldalsosaysupplementaryanglesaretwoanglesthesumofwhosemeasuresisequaltothemeasureofastraightangle.InFig.1.46LaandL{3arePsupplementaryangles.AngleaisthesupplementofL{3;andL{3isthesupplementofLa.B~""Fig.1.43.Fig.1.46.SupPlementaryLS.----------------------- BASICELEMENTSOFGEOMETRY3736FUNDAMENTALSOFCOLLEGEGEOMETRYAtriangleisequilateraliffithasthreecongruentsides.Thepartsofan1.23.Trigangles.Kindsoftriangles.TheunionofthethreesegmentsisoscelestrianglearelabeledinFig.1.49.InthefigureAC==BC.Some-AB,BC,andACiscalledatriangleiffA,B,andCarethreenoncollinearpoints.times,thecongruentsidesarecalledlegsofThesymbolfortriangleisL(plural&,).Thus,inFig.1.47,/':,.ABC=ABUthetriangle.AngleA,oppositeBC,andBCUAC.angleB,oppositeAC,arecalledthebaseanglesoftheisoscelestriangle.SideABistheBMbaseofthetriangle.AngleC,oppositethebase,isthevertexangle.Thesetoftrianglesmayalsobeclassifiedintofoursubsets,accordingtothekindof.JjInteriorBaseangle!anglesthe&,contain(Fig.1.50).AtriangleAisanacutetriangleiffithasthreeacuteangles.Fig.1.49.Isoscelestriangle.AtriangleisanobtusetriangleiffithasoneFig.1.47.Eachofthenoncollinearpointsiscalledavertexofthetriangle,andeachofthelinesegmentsisasideofthetriangle.AngleABC,LACB,andLCABtriangle.InFig.arecalledtheinterioranglesorsimplyt~eanglesof~eI1.47,A,B,andCareverticesofLABC;AB,BC,andCAaresidesofLABC.AngleCisoppositesideAB;ABisoppositeLC.ThesidesACandBCareAcute6Obtuse6saidtoincludeLC.AngleCandLAincludesideCA.ApointPliesintheinteriorofatriangleiffitliesintheinteriorofeachoftheanglesofthetriangle.Everytriangleseparatesthepointsofaplaneinto;:thetriangleitself,thrinteriorofthetriangleandtheexteriorofthetriangle.Theexteriorofatriangleisthesetofpointsoftheplaneofthetrianglethatareneitherelementsofthetrianglenorofitsinterior.Thus,exteriorofLABC=[(interiorofLABC)ULABC]'.ThesetoftrianglesmaybeclassifiedintothreesubsetsbycomparingthesidesoftheL(Fig.1.48).AtriangleisscaleneiffithasnotwosidesthatRight6Equiangular6arecongruent.Atriangleisisoscelesiffithastwosidesthatarecongruent.Fig.1.50.obtuseangle.Atriangleisarighttriangleiffithasonerightangle.Thesidesthatformtherightangleofthetrianglearetermedlegsofthetriangle;candthesideoppositetherightangleiscalledthehypotenuse.InFig.1.51,AB~e<0~seandBCarethelegsandACisthehypot-~-.,(penuseoftherighttriangle.AtriangleisScalene~Isosceles~Equilateral~equiangulariffithasthreecongruentLegA.Bangles.Fig.1.51.Righttriangle.Fig.1.48.-- BASICELEMENTSOFGEOMETRY3938FUNDAMENTALSOFCOLLEGEGEOMETRY14-16.NameapairofcomplementaryanglesineachofthefollowingExercisesdiagrams.I.Usingaprotractorandruler,constructatriangleABCwithmAB=4",17.TellwhyLaandLf3arecomplementaryangles.mLA=110,andmLB=25.cGivetwoname,fmthi,kindzcoftriangle.2.InthefigureforEx.2,whatA'sideiscommonto&ADCandBDC?Whatverticesarecom-ABDmontothetwo&?yABEx.2.x3-12.StatethekindoftriangleEx.14.Ex.15.eachofthefollowingseemstobe(a)accordingtothesidesand(b)accord-ingtotheanglesofthetriangles.(Ifnecessary,usearulertocomparethelengthofthesidesandthesquarecornerofasheetofpapertoccomparetheangles.3.LRST.4.LMNT.ABRsT~IEx.16.Ex.17.RsMNI.18-20.Nameapairofsupplementaryanglesineachofthefollowingdia-Exs.3,4.-~-Iiiiiiiiigures.5.LABC.6.LDEF.DA~R~EAExs.5,6.c7.LGHK.CB8.LABC.A9.LADe.Ex.18.10.LBDC.Ex.19.II.LAEC.A~BD,-D/12.LABE.GH110"lOa"Ex.7.Ex..8-I3.~p~JO80°/~13.InthefigureforExs.8through13,indicatetwopairsofperpendicularABlines.Ex.20.Ex.21. BASICELEMENTSOFGEOMETRY4140FUNDAMENTALSOFCOLLEGEGEOMETRYDc29.ACandEDbisecteachother.A/~~oE---4C-AMBAEx.23.Ex.22.Ex.29.C24.FindthemeasureofthecomplementofeachanglewhosemeasureIS(a)30,(b)45,(c)80,(d)a.25.Findthemeasureofthesupplementofeachanglewhosemeasureis(a)30,(b)45,(c)90,(d)a.30.i5EbisectsLADB.Inexercises26-31,whatconclusionsaboutcongruencecanbedrawnfromthedatagiven?CAEB26.MisthemidpointofAC.~Ex.30.ABBEx.26.\].nisthemidpointofRC.BA027.BDbisectsLABC.Ex.31.A~CDEx.27.1.24.Basingconclusionsonobservationsormeasurements.Ancientmathematiciansoftentestedthetruthorfalsityofastatementbydirectobservationormeasurement.Althoughthisisanimportantmethodofacquiringknowledge,itisnotalwaysareliableone.Letusinthefollowingexamplesattempttoformcertainconclusionsbythemethodofobservationormeasurement.0I.Drawseveraltriangles.Byusingaprotractor,determinethemeasureof28.OCbisectsLACB.eachangleofthetriangles.FindthesumofthemeasuresofthethreeAanglesofeachtriangle.Whatconclusiondoyouthinkyoumightdrawaboutthesumofthemeasuresofthethreeanglesofanygiventriangle?Ex.28. BASICELEMENTSOFGEOMETRY4342FUNDAMENTALSOFCOLLEGEGEOMETRYUnreliableconclusionsbaseduponlimitedorinaccurateobservationsorII-measurementsarecommonalsoinnonmathematicalsituations.For~Fexample,considerthetendencytoassociatesadismwiththepeopleofa\wholenationbecausetheirleadersareguiltyofsadisticcharacteristicsor,ontheotherhand,toattributeglamourtothewomenofagivennationcABbecauseofalimitednumberofcelebratedbeautifulwomeninthehistoryofFig.1.53.Fig.1.52.thatnation.Frequently,theathleticprowessofawholenationisjudgedbytherecordofaverysmallgroupofathletesbelongingtothatnation.2.I~Fig.1.52,LABDand0:BDaresupplementaryadjacentangles.DrawThestudentcanaddmanymoreexamplestothislist.BFbisectingLABDandBEbisectingLCBD.Determinethemeasureof1.25.Theinductivemethodofreasoning.Inthepastexamplesthereason-LEBF.Whatconclusionmightyoudrawfromthisexperiment?ingwhichwasusedinarrivingatconclusionsisknownasinductivereasoning.3.DrawtwointersectinglinesasinFig.1.53.MeasureLaandLf3.AlsoAgeneralconclusionisdrawnbyinvestigatinganumberofparticularcases.measureLaandLc:f>.Givepossibleconclusionsaboutverticalangles.Itisthemethodofresearch.Inductivereasoninghasmadealargecontribu-IftheanglesofthetrianglesofExample1weremeasuredcarefully,thetiontocivilization.Initoneobserves,measures,studiesrelations,computes,studentwilldiscoverthatthesumofthemeasuresofthethreeanglesofanyanddrawsconclusions.Thesetentativeconclusionsarecalledhypotheses.trianglewillalwaysbeIleal'180.Isthestudent,asaresultofsuchmeasure-Wewillusemanyhypothesesinthistext.Thehypothesisindicatesastate-ments,justifiedinstatingunequivocallythatthesumofthemeasuresofthementthatispossiblytruebasedonobservationofalimitednumberofcases.threeanglesofanytriangleis180?ThefinerthemeasuringinstrumentsandthemorecarefultheobservationsLetusconsidertheimplicationsofmakingsuchaconclusion.Firsttheandmeasurements,thegreaterthepossibilitythatthehypothesisiscorrect.triangleshadtobedrawninordertomeasuretheangles.Thewidthofthe:'ationalpre-electionpollsareconductedbyobservingagoodrepresentativelinesrepresentingthesidesofthesetriangleswillvarydependinguponthecrosssectionofthevariousregionsofthenation.Expertshavebeenabletofinenessofthedrawinginstrument.TheprotractorwithwhichtheanglesmakeveryaccuratepredictionsbyobservinglessthaninpercentofallthearemeasuredisroughlydIvIdedultodegreesonly.TllUsti,Cprotractoreligiblevotersinanationalelection.couldnotshowadiffep'na'of/gofadegreethatmightexistbetweenthesum1.26.Thedeductivemethodofreasoning.Inductivereasoningproceedsofthemeasuresoftheanglesoftwotriangles.Nomatterhowfinethesidesbyobservingaspecificcommonpropertyinalimitednumberofcasesandofthetrianglemaybedrawnorhowaccuratethemeasuringinstrument,concludingthatthispropertyisgeneralforallcases.Thus,itproceedsfromtherewillalwaysbeapossibilitythat,iftheaccuracyofthemeasurementsthespecifictothegeneral.However,atheorymayholdforseveralthousandwereincreased,aslighterrorintheanglesummightbedetected.casesandthenfailontheverynextone.WecanneverbeabsolutelycertainAsecondfallacyinstatingasanabsolutetruththesumofthemeasuresofthatconclusionsbaseduponinductivereasoningarealwaystrue.theanglesofanytriangleis180istheassumptionthatwhatmaybetrueforaAmoreconvincingandpowerfulmethodofdrawingconclusionsiscalledlimitednumberofcasesmustbetrueforallcases.Thisisanunreliabledeductivereasoning.Whenreasoningdeductively,oneproceedsfromthepractice.Wewouldbesaferinstatingthattheresultsofourexperienceleadgeneraltothespecific.Onestartswithalimitednumberofgenerallyac-ustobelievethatprobablytheanglesumofanytriangleequals180.ceptedbasicassumptionsandbyabuildingprocessoflogicalstepsprovesotherInlikemannerwewouldbejustifiedinstatinginExample2thatitappearsfacts.Thus,wemaybuildupontheseacceptedassumptionsandderivedthattheanglebisectorsoftwoadjacentsupplementaryanglesareperpen-factsinamannerthatwillenableuseventuallytoprovethedesiredconclusion.diculartoeachother.InExample3,wecouldstatethatitappearsthattheTheseprovedfactsaretermedtheorems.pairsofthenonadjacentanglesformed,whentwolinesintersect,arecon-A.lldeductivereasoninginvolvesacceptanceofthetruthofacertainstate-gruent.ment(orstatements),calledanassumption.ThisassumptionneednotbeInsubsequentstudyinthistextwewillprovethateachoftheaboveapparentobvioustothereadernorneeditbeagenerallyacceptedfact,butitmustconclusionsaretruthsinfact,but,untilwedoprovethem,wecanonlystatebeacceptedforthepurposeofprovingaparticularargument.Changingwhatseemstobetrue.--------- BASICELEMENTSOFGEOMETRY4544FUNDAMENTALSOFCOLLEGEGEOMETRYMr.SmithwasborninthecityofCarpenteria.Carpenteriaisinthethebasicassumptionswillgenerallyaltertheresultantconclusions.InUnitedStates.attemptingtoproveaparticularargument,theoriginallyacceptedassump-8.Allquadrilateralshavefoursides.Arhombushasfoursides.tionmayleadtoacontradictionofotheracceptedassumptionsorofother9.Onlystudentswhostudyregularlywillpassgeometry.BillSmithdoesprovedfacts.Inthisevent,thetruthoftheoriginalassumptionmustbenotstudyregularly.questioned;orpossibly,thetruthoftheacceptedassumptionsmaythenbe10.MaryisinanEnglishclass.Allfreshmenincollegeareenrolledinsomedoubted.Englishclass.Whenacertainassumptionisaccepted,certainconclusionsinevitablyII.BaseballplayerseatZeppocerealandarealertonthediamond.Ieatfollow.TheseconclusionsmaybefalseiftheassumptionsonwhichtheyareZeppocereal.basedarefalse.Itisimperative,then,thatwedistinguishbetweenvalidity12.Thefirst-andthird-periodgeometryclassesweregiventhesametest.andtruth.Considerthefollowingstatements:(1)Allmenarebrave.(2)Studentsinthefirst-periodclassdidbetterthanthoseinthethird-FrancisJonesisaman.(3)FrancisJonesisbrave.Statement3isavalidperiodclass.Dickwasenrolledinthefirst-periodclassandStanwasinconclusionofassumptions1and2,butitneednotbetrue.Ifeitherstate-thethird-periodclass.ment1orstatement2isfalse,itispossiblethatstatement3isalsonottrue.Itisnecessaryinseekingthetruthofconclusionsthatthetruthofthebasic13-22.Answerthefollowingquestionstocheckyourreadingandreason-premisesuponwhichtheyarebasedbeconsideredcarefully.ingability.Bothinductionanddeductionarevaluablemethodsofreasoninginthe13.Whycan'taman,livinginWinston-Salem,beburiedwestoftheMissis-studyofgeometry.NewgeometrictruthscanbediscoveredbyinductivesippiRiver?Ieasoning.D~cillctiv~J~,lsO_l1inK(:a!1uthenbeusedinprovingthatsuch----14.-~()me:m(WitnsTia-ve:)Udays,somehave:)1days.HowmanynavE2S-aafsTdiscoveriesaretrue.15.IhaveinmyhandtwoU.S.coinswhichtotal55cents.OneisnotaAftertrying,inthenextexercise,ourskillatdeductivereasoning,wewillnickel.Placethatinmind.Whatarethetwocoins?studyatgreaterlengthinChapter2whatconstitutes"logical"reasoning.16.Afarmerhad17sheep.Allbut9died.Howmanydidhehaveleft?Wewillthenbebetterpreparedtorecognizewhenwehaveprovedour17.Twomenplaycheckers.Theyplayfivegamesandeachmanwinsthetheorems.Thestudentshouldnotbetooconcernedatthisstageifhefail~samenumberofgames.Howdoyoufigurethatout?togivecorrectanswerstothefollowingexercise.18.Ifyouhadonlyonematchandenteredaroomwheretherewasalamp,an011heater,andsomekmdlmgwood,whIChwouldyoulIghthrst?Exercises(A)19.Taketwoapplesfromthreeapplesandwhatdoyouhave?Inthefollowingexercisessupplyavalidconclusionifonecanbesupplied.20.IsitlegalinNorthCarolinaforamantomarryhiswidow'ssister?Ifnoconclusionisevident,explainwhy.21.Thearchaeologistwhosaidhefoundagoldcoinmarked46B.C.waseitherlyingorkidding.Why?1.Mrs.jones'dogbarkswheneverastrangerentersheryard.Mrs.Jones'22.Awomangivesabeggar50cents.Thewomanisthebeggar'ssister,dogisbarking.butthebeggarisnotthewoman'sbrother.Howisthispossible?2.Waterinthefishpondfreezeswheneverthetemperatureisbelow32°Fahrenheit.Thetemperaturebythefishpondis30°Fahrenheit.Exercises(B)3.Allcollegefreshmanstudentsmusttakeanorientationclass.MarySmithisafreshmancollegestudent.Eachofthefollowingexercisesincludeafalseassumption.Disregardthe4.Allmembersofthebasketballteamaremorethan6feettall.Lllsityoftheassumptionandwritetheconclusionwhichyouarethenforcedismorethan6feettall.toaccept.5.Collegestudentswillbeadmittedtothebaseballgamefree.1.Giventwomen,thetallermanistheheavier.BobistallerthanJack.Brownwasadmittedtothebaseballgamefree.2.Barkingdogsdonotbite.Mydogbarks.6.Tim'sdadalwaysbuyscandywhenhegoestothedrugstore.3.Whenapersonwalksunderaladder,misfortunewillbefallhim.Mr.Tim'sdadboughtsomecandy.Grimeswalkedunderaladderyesterday.7.AnypersonbornintheUnitedStatesisacitizenoftheUnitedStates. 4.Allwomenarepoordrivers.JerryWallaceisawoman.5.Anyonehandlingatoadwillgetwartsonhishand.Ihandledatoadtoday.6.Oftwopackages,themoreexpensiveisthesmaller.Mary'sChristmaspresentwaslargerthanRuth's.Inthefollowingexercise,indicatewhichofthefollowingconclusionslogicallyfollowfromthegivenassumptions.7.Assumption:AllmembersoftheOogatribearedark-skinned.Nodark-skinnedpersonhasblueeyes.Conclusion:(a)1:0Oogatribesmanhasblueeyes.SummaryTests(b)Somedark-skinnedtribesmenaremembersoftheOogatribe.(c)Somepeoplewithblueeyesarenotdark-skinned.(d)SomeOogatribesmenhaveblueeyes.8.Assumption:Onlyoutstandingstudentsgetscholarships.Allout-standingstudentsgetpublicity.Test1Conclusion:Indicnetheonewordornumberthatwillmakethefollowingstatements(a)Allstudentswhogetpublicitygetscholarships.xue.(b)Allstudentswhogetscholarshipsgetpublicity.(c)Onlystudentswithpublicitygetscholarships.I.Thesidesofarightanglearetoeachother.(d)Somestudentswhodonotgetpublicitygetscholarships.2.Thepairsofnonadjacentangles[ormedwhentwolinesintersectare9.Assumption:Somecookedvegetablesaretasty.Allcookedvegetable~calledarenourishing.3.Anangleislargerthanitssupplement.Conclusiun:4.Thesideoppositetherightangleofatriangleiscalledthe(a)Somevegetablesaretasty..J.Atrianglewithnotwosidescongruentiscalledatriangle.(b)Ifavegetableisnotnourishing,itisnotacookedvegetable.6.Ifthesumofthemeasuresoftwoanglesis180,theanglesare-(c)Sometastyvegetablesarenotcooked.7.Atrianglewithtwocongruentsidesiscalledatriangle.(d)Ifavegetableisnotacookedvegetable,itisnotnourishing.8.V3isa(n)realnumber.9.Theofanangledividestheangleintotwoangleswithequalmeasures.10.ComplementaryanglesaretwoanglesthesumofwhosemeasuresisequaltoII.Thedifferencebetweenthemeasuresofthecomplementandthesupple-mentofanangleisalways12.isthenon-negativeintegerthatisnotacountingnumber.13.Thesumofthemeasuresoftheanglesaboutapointisequalto14.Theanglewhosemeasureequalsthatofitssupplementisaangle.15.Ananglewithameasurelessthan90is16.Angleswiththesamemeasuresare-17.Theintersectionoftwodistinctplanesiseitheranullsetora18.Anangleisthe46~-oftworayswhichhaveacommonendpoint.47 48FUNDAMENTALSOFCOLLEGEGEOMETRYSUMMARYTESTS4919.Theonlypointofalineequallydistantfromtwoofitspointsisthe17.Ifa+h<0,thenla+hl,-121-<_3212-2-102"'3Probs.1-8.1.mDH=2.mCB=3.mDF=4.mHB=5.mce=6.mCF=7.ThecoordinateofthemidpointofEHis8.ThecoordinateofthemidpointofDHis9.IfmLAis40,whatisthemeasureofthecomplementofLA?10.IfmLBis110,whatisthemeasureofthesupplementofLB?and0fJbi-ElementaryLogic11-16.Given:mLAOC=40;mLCOE=70;DBbisectsLAOCsectsLCOE.Completethefollowing.11.mLAOB=12.mLCOD=2.1.Logicalreasoning.Wehaveallheardthewords"logic"and"logical"Iused.Wespeakofaperson'sactionasbeing"logical,"orofa"logical"113.mLBOD=14.mLEOB=solutiontoaproblemA"logical"behaviorisa"reasonable"behavior.The1.1').mLAOE=16.mLBOE="illogical"conclusionisan"unreasonable"conclusion.Whenapersonengagesin"clearthinking"or"rigorousthinking,"heisemployingthedisciplineoflogicalreasoning.Probs.11-16.AInthischapterwewilldiscussthemeaningsofafewwordsandsymbolsusedinpresent-daylogicandmathematics.Wewillthenintroducesomeofthe17.JBandtI5arestraightlinesintersectingatE.Whatmustthemeasuresmethodsandprinciplesusedindistinguishingcorrectfromincorrectargument.Wewillsystematizesomeofthesimplerprinciplesofvalidreasoning.ofanglesa:ande/>be?18.Themeasureofangle()isthreetimesthemeasureofLe/>.WhatistheAlthoughthemethodofdeductivelogicpermeatesallfieldsofhumanknowledge,itisprobablyfoundinitssharpestandclearestforminthestudymeasureofLe/>?ofmathematics.cMB2.2.Statements.Adiscourseiscarriedonbyusingsentences.Someofthesesentencesareintheformofstatements.Vq)Definition:Astatementisasentencewhichiseithertrueorfalse,butnotSeboth.~TAProb.17.Prob.18.Itshouldbenotedherethatthewords"true"and"false"areundefinedCelements.Everystatementisasentence;butnoteverysentenceisastate-19.CMbisectsLACB;mLACB=110.ment.AstatementissaidtohaveatruthvalueTifitistrueandFifitisfalse.ThenmLBCM=-.Suchthingsasaffirmations,denials,reports,opinions,remarks,comments,20.Anglea:isthecomplementofanandjudgmentsarestatements.Everystatementisanassertion.anglewhosemeasureis38;L{3isA~8Thesentence"SanFranciscoisinCalifornia"isastatementwithatruththesupplementofLa:.ThenMvalueT.Thesentence"Everynumberisodd"isastatementwithatruthmL{3=Prob.19.valueF.5051 ELEMENTARYLOGIC53FU!'DAMENTALSOFCOLLEGEGEOMETRY1.ItishotandIamtired.Allstatementsinthefieldoflogicareeithersimplesentencesorcompound2.BaseballplayerseatZeppocerealandarealertonthediamond.mtences.Thesimplesentencecontainsonegrammaticallyindependent3.Hisactionwaseitherdeliberateorcareless.tatement.Itdoesnotcontainconnectingwordssuchasand,or,andbut.A4.ThecomposerwaseitherChopinorBrahms.mpoundsentenceisformedbytwoormoreclausesthatactasindependent5.Thefigureisneitherasquarenorarectangle...then,ifandntencesandarejoinedbyconnectivessuchasand,or,but,if.6.EitherJonesisinnocentorheislying.lyif:either...or,andneither...nor.7.HeiscleverandIamnot.8.SueandKayarepretty.Examples9.SueandKaydislikeeachother..Everynaturalnumberisoddoreven.10.Thatanimaliseitherdeadoralive..Iamgoingtocashthischeckandbuymyselfanewsuit.II.Twolineseitherintersectortheyareparallel..ThewindisblowingandIamcold.12.Ifthisobjectisneitheramalenorafemale,itisnotananimal..IwillgototheshowifJohnasksme.13.Everyanimaliseitheramaleorafemale.'.Peoplewhodonotworkshouldnoteat.14.Thecostisneithercheapnorexpensive.Itiscustomaryinlogictorepresentsimplestatementsbylettersasp,q,r,15.Iwouldbuythecar,butitcoststoomuch.:tc.Henceifweletpindicatethestatement,"Thewindisblowing"and16.Asquareisarectangle.indicate,"Iamcold,"wecanabbreviateStatement3aboveaspandq.2.3.COl;ljunction.Wehaveseenhowtwostatementscanbeconnectedtomakeanotherstatement.Someoftheseformsoccurrepeatedlyinlogical~xercises(A)discourseandareindispensableforpurposesofanalysis.WewilldefineandConsiderthefollowingsentences.Whicharestatements?discusssomeofthemorecommononesinthischapter.1.Howmanyarethere?Definition:Ifpandqarestatements,thestatementoftheformpandqis2.3plus2equals5.calledtheconjunctionofpandq.Thesymbolforpandqis"pIIq."3.;)X2equals5.Therearemanyotherwordsinordinaryspeechbesides"and,"thatare4.Givemethetext.usedasconjunctives;e.g.,"but,""although,""however,""nevertheless."5.TomisolderthanBill.6.Allrightangleshavethesamemeasure.ExamPlesI.Itisdaytime;however,Icannotseethesun.7.Sheishungry.2.Iamstarved,butheiswellfed.8.Mrs.Jonesisill.3.MaryisgoingwithGeorgeandRuthisgoingwithBill.9.Heisthemostpopularboyinschool.4.Somerosesareredandsomerosesareblue.10.IfIdonotstudy,Iwillfailthiscourse.5.SomerosesareredandtodayisTuesday.11.IfIliveinLosAngeles,IliveinCalifornia.12.xplus3equals5.Althoughthedefinitionforconjunctionseemssimpleenough,weshould13.Goaway!notacceptitblindly.Youwillnotethatourdefinitiontakesforgrantedthat14.Thewindowisnotclosed."pandq"willalwaysbeastatement.Rememberasentenceisnotastatement15.3X2doesnotequalS.unlessitiseithertrueorfalse,butnotboth.Itbecomesnecessary,then,to16.Howmuchdoyouweigh?formulatesomerulewhichwecanusetodeterminewhen"pandq"istrueandwhenitisfalse.Withoutsucharule,ourdefinitionwillhavenomeaning.Exercises(B)Eachofthefollowingstatementsisintheformof"pandq."Checkwhichonesaretrueandwhichonesarefalse,andthentrytoformulateaIneachofthefollowingexercisesthereisacompoundstatementoronegeneralrulefordecidinguponthetruthsofaconjunction.thatcanbeinterpretedasone.Statethesimplecomponentsofeachsentence.-------- 54FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGIC1.2+3=5and2X3=5.qarebothfalse.Itwillberecalledthatweinterpreted"or"intheinclwsenseinourdefinitionoftheunionofsets.2.2isanevennumberand3isanoddnumber.3.2isanevennumberand4isanevennumber.Definition:Thedisjunctionoftwostatementspandqisthecompm:4.2isanoddnumberand4isanoddnumber.sentence"porq."Itisfalsewhenbothpandqarefalseandtruein5.Acirclehastensidesandatriangle0,0hasthreesides.othercases.Thesymbolfortheinclusiveporqis"pVq."~-+-6..113U{A}=A13and,113U{A,B}=A13.Thetruthtableforthedisjunction"porq"follows:Instudyingtheforegoingexamples,youshouldhavediscoveredthat"Pandq"isconsideredtrueonlywhenbothpandqaretrue.IfeitherpisIJqIJVqfalseorqisfalse(orbotharefalse),then"pandq"isfalse.Thisissome-TTTtimesshownmostclearlybythetruthtablebelow.TFTpqP/qFTTFFFTTTTFFExercisesFTFIneachofthefollowingexercisestherearetwostatements.jointheFFFstatementsfirsttoformaconjunctionandthentoformadisjunction.Detlminethetruthorfalsityofeachofthecompoundsentences."pandq"istrue(T)inonlyonecaseandfalse(F)inallothers.Itshould1.Thediamondishard.Puttyissoft.beemphasizedthattruthtablescannotbeproved.Theyrepresentagree-2.Thestatementistrue.Thestatementisfalse.mentsintruthvaluesofstatementsthathaveprovedusefultomathematicians3.Thetwolinesintersect.Thelinesareparallel.andlogicians.4.Arayisahalf-line.Araycontainsavertex.2.4.Disjunction.Anotherwaytocombinestatementsisbyusingthe5.Thereare30daysinFebruary.Fiveislessthan4.connective"or"betweenthem.Considerthefollowingsentences:6.~()trianglehasfoursides.Asquarehasfoul'sides.7.three"pluszeroequa1S3:ThreetImes-zeroequals:r:------1.Iplantogotothe-game-ortotJieilioW.~--~--~~~~~-H.Someanimalsaredogs.Somedogsbark.2.IexpecttoseejohnorTomattheparty.9.AisintheinteriorofLA13C.CisonsideABofLABC.3.Themusicteachertoldmysonthathecoulddowellasastudentofthe10.Allwomenarepoordrivers.MynameisMudd.pianoortheflute.11.Thesunishot.Dogscanfly.Inthefirstsentenceitisclearthatthespeakerwillgoeithertothegameor12.-5islessthan2.4ismorethan3.totheshowbutthathewillnotdoboth.Itisnotclearinthesecondsentence13.Anangleisformedbytworays.Anintervalincludesitsendpoints.ifthespeakerwillseeonlyjohnoronlyTomattheparty.Itmightmean14.LA13CnLABC=LA13C.LA13CULA13C=LABC.thathewillseeboth.Inthethirdsentence,itshouldbeclearthattheson15.Thesidesofanangleisnotasubsetoftheinterioroftheangle.Chri~shoulddowellwitheitherorbothinstruments.masoccursinDecember.Thusweseethatthecommonuseoftheword"or"oftenleadstoambiguity16.Thesupplementofanangleislargerthanthecomplementoftheanglandnotuniformmeaning.SometimesitindicatesonlyoneofthestatementsThemeasureofanacuteangleisgreaterthanthemeasureofanobtuwhichmakeupthedisjunctionistrue.Sometimesitisusedtomeanatleastangle.oneofthestatementsandpossiblybotharetrue.Inlogicwecannottolerate2.5.Negation.Statementscanbemadeaboutotherstatements.One,suchvariedmeanings.Wemustagreeonpreciselywhatwemeanwhenwe"thesimplestandmostusefulstatementofthistypehastheform"pisfalsesay"porq."Mathematicianshaveagreedthat,unlessitisexplicitlystated1Everyonehasprobablymadeastatementthathebelievedtrueonlytoha1tothecontrary,theconnective"or"shouldbeusedintheinclusivesense.I'1someoneelseshowhisdisagreementbysaying,"Thatisnottrue."Thusstatementsoftheform"porq"aretrueinallcasesexceptwhenpand! 56FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGDefinition:Thenegationofastatement"P"isthestatement"not-p."It8.Itisnottruethat2plusfourequals6.means"pisfalse";or"itisnottruethatp."Thesymbolfornot-pis"~p."9.Twoplus4equals8.Thenegationofastatement,however,isnotusuallyformedbyplacinga10.Perpendicularlinesformrightangles.I].Allequilateraltrianglesareequiangular."not"infrontofit.Thisusuallywouldmakethesentencesoundawkward.Thuswherepsymbolizesthestatement,"Allmisersareselfish,"thevarious12.Allblindmencannotsee.statements,"Itisfalsethatallmisersareselfish,""Notallmisersareselfish,"13.Someblindmencarrywhitecanes."Somemisersarenotselfish,""Itisnottruethatallmisersareselfish"are14.Allsquaresarerectangles.symbolized"not-p."Thenegationofanytruestatementisfalse,andthe15.Allthesecookiesaredelicious.negationofanyfalsestatementistrue.Thisfactcanbeexpressedbythe]6.Someofthestudentsaresmarterthanothers.truthtable.17.EveryEuropeanlivesinEurope.18.Foreveryquestionthereisananswer.p~/}]9.Thereisatleastonegirlintheclass.20.Everyplayeris6feettall.TF21.Somequestionscannotbeanswered.FT22.Somedogsaregreen.23.EveryZEPisaZOP.24.Somepillowsaresoft.Indevelopinglogicalproofs,itisfrequentlynecessarytostatethenegation25.Anullsetisasubsetofitself.ofstatementslike"Allfatpeoplearehappy"and"Somefatpeopleare26.l'oteveryangleisacute.happy."Itshouldbeclearthat,ifwecanfindoneunhappyfatperson,wewillhaveprovedthefirststatementtobefalse.Thuswecouldformthe2.6.Negationsofconjunctionsanddisjunctions.Indeterminingthenegationbystating"Somefatpeoplearenothappy"or"Thereisatleastofthenegationofaconjunctionoradisjunctionweshouldfirstrecallonefatpersonwhoisnothappy."Butwecouldnotformthenegationby.whatconditionsthecompoundsentencesaretrue.Toformthenegati,thestatement"Nofatpersonishappy."Thisisacommonerrormadeby"Achickenisafowlandacatisafeline,"wemustsaythestatementistheloosethinker.Thenegationof"allare"is"somearenot"or"notallWecandothisbystatingthatatleastoneofthesimplestatementsisare."Vecandothisbystating"AchickenisnotafowloracatisnotafelTheword"some"incommonusagemeans"morethanone."However,Thenegationof"IwillstudybothSpanishandFrench"couldbe"Iwilinlogicitwillbemoreconvenientifweagreeittomean"oneormore."ThisstudySpanishorIwillnotstudyFrench."wewilldointhistext.ThusthenegationofthesecondstatementaboveItshouldbeclearthatthenegationof"pandq"isthestatement"notwouldbe"Nofatpersonishappy"or"Everyfatpersonisunhappy."Thenot-q."Intruthtableform:negationof"someare"is"noneare"or"itisnottruethatsomeare."/}qq~(Plq)ExercisesP1~/}~q~pV~qIneachofthefollowingformthenegationofthestatement.TTTFFFFI.Goldisnotheavy.TFFTFTT2.Fidoneverbarks.FTFTTFTFFFTTTT3.Anyonewhowantsagoodgradeinthiscoursemuststudyhard.4.Aspirinrelievespain.5.Ahexagonhassevensides.6.Itisfalsethatatrianglehasfoursides.Toformthenegationofthedisjunction,"Wearegoingtowinor7.Noteverybankerisrich.informationisincorrect"wewrite,"WearenotgoingtowinandmyinfO!L 58FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGICtioniscorrect."Thusthenegationof"porq"is"not-(porq),"andthisI.If5x=20,thenx=4.means"notpandnotq."Intruthtableform:2.Ifthisfigureisarectangle,thenitisaparallelogram.pqpVq~(pVq)~p~q~pl~qAhypotheticalstatementassertsthatitsantecedentimPliesitsconsequelThestatementdoesnotassertthattheantecedentistrue,butonlythattiTTTFFFFconsequentistrueiftheantecedentistrue.TFTFFTFItiscustomaryinlogictorepresentstatementsbyletters.Thus,wemigFTTFTFFletprepresentthestatement,"Thefigureisarectangle"andqthestatemeIFFFTTTT"Thefigureisaparallelogram."Wecouldthenstate,"Ifp,thenq"orimpliesq."Weshallfinditusefultouseanarrowfor"implies.'canwrite"p~q."SuchastatementiscalledanimPlication."WethtExercisesThe"if"statementdoesnothavetocomeatthebeginningofthecanGivethenegationofeachstatementbelowanddetermineifitistrueorpoundstatement.Itmaycomelast.Inothercases,thepremisewill114false.startwiththeword"if."Forexample:I.Anapricotisafruitandacarrotisavegetable.I.Agoodscoutistrustworthy.2.LincolnwasassassinatedorDouglasswasassassinated.2.Applesarenotvegetables.3.Somemenliketohunt,othersliketofish.3.Thestudentinthisclasswhodoesnotstudymayexpecttofail.4.Somenumbersareoddandsomeareeven.5.Nonumbersareoddandallnumbersareeven.Eachoftheabovecanbearrangedtothe"if-then"formasfollows:6.Alllinesaresetsofpointsorallanglesarerightangles.7.ThesidesofarightangleareperpendicularandallrightanglesareI.Ifheisagoodscout,thenheistrustworthy.congruent.2.Ifthisisanapple,thenitisnotavegetable.8.Theintersectionoftwoparallellinesisanullsetoreachpairofstraight3.Ifthestudentinthisclassdoesnotstudy,thenhewillfail.lineshasapointcommontothetwolines.9.Everytrianglehasarightangleandanacuteangle.Otheridiomsthathavethesamemeaningas"ifp,thenq"are:"ponlyifq,'10.Everytrianglehasarightangleandanobtuseangle."pisasufficientconditionforq,""q,ifp,""q,isanecessaryconditionforp,'II.Everytrianglehasarightangleoranobtuseangle."wheneverp,theng,""supposep,thenq."12.Notrianglehastwoobtuseanglesortworightangles.Supposeyourinstructormadethestatement,"IfyouhandinallyoUl13.Sometriangleshavethreeacuteanglesandsomehaveonlytwoacutehomework,youwillpassthiscourse."Herewecanletprepresentthestate.angles.ment,"Youhandinallyourhomework,"andgthestatement,"Youwillpas~14.AidesignatesalineandAlfdesignatesaray.thecourse."Ifbothpandgaretrue,thenp~qiscertainlytrue.Suppose15.Arayhasoneend-pointorasegmenthastwoend-points.pistrueandqisfalse;i.e.,youhandinallyourhomeworkbutstillfailthecourse.Obviously,then,p~qisfalse.2.7.Logicalimplication.Themostcommonconnectiveinlogicaldeduc-:Textsupposepisfalse.Howshallwecompletethetruthtable?Ifpistionis"if-then."Allmathematicalproofsemployconditionalstatementsof;,falseandqistrue,youdonothandinallyourhomeworkbutyoustillpassthethistype.Theifclause,calledhypothesisorpremiseorgivenisasetofoneor,jcourse.Ifpisfalseandqisfalse,youdonothandinallyourhomeworkmorestatementswhichwillformthebasisforaconclusion.Thethenclauseandyoudonotpassthecourse.Atfirstthoughtonemightfeelthatnotruthwhichfollowsnecessarilyfromthepremisesiscalledtheconclusion.Thevalueshouldbegiventosuchcompoundstatementunderthoseconditions.statementimmediatelyfollowingthe"if"isalsocalledtheantecedent,andtheIfWedidso,wewouldviolatethepropertythatastatementmustbeeitherstatementimmediatelyfollowingthe"then"istheconsequent.trueorfalse.Herearesomesimpleexamplesofsuchconditionalsentences:Logicianshavemadethecompletelyarbitrarydecisionthatp~qistrue ELEMENTARYLOGIC6160FUNDAMENTALSOFCOLLEGEGEOMETRY~qisconsideredInapplyingtheRuleofInference,itdoesnotmatterwhatthecontentofthewhenpisfalse,regardlessofthetruthvalueofq.Thus,Pstatementspandqare.Solongas"pimpliesq"istrueandpistrue,welogic-falseonlyifpistrueandqisfalse.Thetruthtableforp~qis:allymustconcludethatqistrue.Thisisshownbyformingthegeneralstructure:pqp-7q1.I)~Ifor1.f)~q.2.ETTT~3.:.qTFF3.:.qFTTFFTThesymbol:.means"then"or"therefore."Thethree-stepformiscalledasyllogism.StepsIand2arecalledtheassumptionsorpremises,andstep3iscalledtheconclusion.TheorderofthestepsIand2canbereversedandExercisesnotchangethevalidityofthesyllogism.ThusthesyllogismcouldalsobeIneachofthefollowingcompoundsentencesindicatethepremiseandthewritten:conclusion.1.Por1.p.2.P~qI.Thetrainwillbelateifitsnows.2.p~q3.:.q2.ApersonlivesinCaliforniaifhelivesinSanFrancisco.3.:.If3.Onlycitizensover21havetherighttovote.4.Fourislargerthanthree.Acommontypeofinvalidreasoningisthatofaffirmingtheconsequent.Its5.Allstudentsmusttakeaphysicalexamination.structurefollows:6.IknowhewastherebecauseIsawhim.7.Twolineswhicharenotparallelintersect.1.1)§>18.Allrightanglesarecongruent.9.Naturalnumbersareeitherevenorodd.~~.J.:.plOHewillbepunishedifheiscaught.11.Everyparallelogramisaquadrilateral.2.9.Modustollens.Asecondsyllogismdeniestheconsequentofanin-12.Goodscoutsobeythelaws.ferenceandthenconcludestheantecedentoftheconditionalsentencemust13.Birdsdonothavefourfeet.bedenied.Thismodeofreasoningiscalledmodustollens.Modustollem14.Diamondsareexpensive.reasoningcanbestructured:15.Thosewhostudywillpassthiscourse.16.Thesidesofanequilateraltrianglearecongruenttoeachother.I.p~q17.Thepersonwhostealswillsurelybecaught.2.not-q18.Tobesuccessful,onemustwork.19.Theworke-rwillbeasuccess.3.:.not-f)20.Youmustbesatisfiedoryourmoneywillberefunded.Considertheconditionalsentence(a)"Ifitisraining,itiscloudy."Then21.Withyourlooks,I'dbeamoviestar.considerwiththeinferencethestatement(b)"Itisnotcloudy."Ifpremises2.8.Modusponens.Animplicationbyitselfisoflittlevalue.However,if(a)and(b)hold,wemustconcludebymodustollensreasoningthat(c)"Itisnotweknow"pimpliesq"andthatpisalsotrue,wemustacceptqastrue.Thisraining."isknownastheFundamentalRuleofInference.ThisruleofreasoningiscalledThemethodofmodustollensisalogicalresultoftheinterpretationthatmodusponens.Forexample,considertheimplication:(a)"Ifitisraining,itisI)~qmeans"qisanecessaryconditionforp."Thus,ifwedon'thaveq,wecloudy."Also,withtheimplicationconsiderthestatement(b)"Itisraining."can'thavep.Ifweaccept(a)and(b)together,wemustconcludethat(c)"Itiscloudy." ELEMENTARYLOGIC6150FUNDAMENTALSOFCOLLEGEGEOMETRY~qisconsideredInapplyingtheRuleofInference,itdoesnotmatterwhatthecontentofthewhenpisfalse,regardlessofthetruthvalueofq.Thus,Pstatementspandqare.Solongas"pimpliesq"istrueandpistrue,welogic-falseonlyifpistrueandqisfalse.Thetruthtableforp~qis:allymustconcludethatqistrue.Thisisshownbyformingthegeneralstructure:pqp~q1.jJ~qor1.fJ~q.2.12TTT~3.:.qTFF3.:.qFTTFFTThesymbol:.means"then"or"therefore."Thethree-stepformiscalledasyllogism.Steps1and2arecalledtheassumptionsorpremises,andstep3iscalledtheconclusion.Theorderofthesteps1and2canbereversedandExercisesnotchangethevalidityofthesyllogism.ThusthesyllogismcouldalsobeIneachofthefollowingcompoundsentencesindicatethepremiseandthewritten:conclusion.1.Por1.p.2.P~qII.Thetrainwillbelateifitsnows.2.p~q3.:.q:2.ApersonlivesinCaliforniaifhelivesinSanFrancisco.3.:.q3.Onlycitizensover21havetherighttovote.4.Fourislargerthanthree.Acommontypeofinvalidreasoningisthatofaffirmingtheconsequent.Its5.Allstudentsmusttakeaphysicalexamination.structurefollows:6.IknowhewastherebecauseIsawhim.I7.Twolineswhicharenotparallelintersect.1.p~8.Allrightanglesarecongruent.2~9.Naturalnumbersareeitherevenorodd.'t:.f;10.Hewillbepunishedifheiscaught.11.Everyparallelogramisaquadrilateral.2.9.Modustollens.Asecondsyllogismdeniestheconsequentofanin-12.Goodscoutsobeythelaws.ferenceandthenconcludestheantecedentoftheconditionalsentencemust13.Birdsdonothavefourfeet.bedenied.Thismodeofreasoningiscalledmodustollens.Modustollens14.Diamondsareexpensive.reasoningcanbestructured:15.Thosewhostudywillpassthiscourse.16.Thesidesofanequilateraltrianglearecongruenttoeachother.I.p~q17.Thepersonwhostealswillsurelybecaught.2.not-q18.Tobesuccessful,onemustwork.19.Theworke-rwillbeasuccess.3.:.not-p20.Youmustbesatisfiedoryourmoneywillberefunded.Considertheconditionalsentence(a)"Ifitisraining,itiscloudy."Then21.Withyourlooks,I'dbeamoviestar.considerwiththeinferencethestatement(b)"Itisnotcloudy."Ifpremises2.8.Modusponens.Animplicationbyitselfisoflittlevalue.However,if(a)and(b)hold,wemustconcludebymodustollensreasoningthat(c)"Itisnotweknow"pimpliesq"andthatpisalsotrue,wemustacceptqastrue.Thisraining."isknownastheFundamentalRuleofInference.ThisruleofreasoningiscalledThemethodofmodustollensisalogicalresultoftheinterpretationthatmodusponens.Forexample,considertheimplication:(a)"Ifitisraining,itisf;~qmeans"qisanecessaryconditionforp."Thus,ifwedon'thaveq,wecloudy."Also,withtheimplicationconsiderthestatement(b)"Itisraining."can'thavep.Ifweaccept(a)and(b)together,wemustconcludethat(c)"Itiscloudy."~-~_. 62FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGIC63Anothercommontypeofinvalidreasoningisthatofdenyingtheantecedent.1.Thetalleroftwomenisalwaystheheavier.BobistallerthanJack.Itsstructurefollows:2.Allquadrilateralshavefoursides.Arhombushasfoursides.3.Barkingdogsdonotbite.Mydogbarks.1.p-;(JJ4.TriangleABCisequilateral.Equilateraltrianglesareisosceles.5.Everyparallelogramisaquadrilateral.FigureABCDisaparallelogram.f~~~-q6.IfBEAC,thenmAB+mCB=mAC.BEAC.7.lfa=b,thena+c=b+c.a=b.8.Ifa=b,thenc=d.c=d.Twootherprinciplesoflogicshouldbementionedhere.TheLawofthe9.Parallellinesdonotmeet.LinesIandmdonotmeet.ExcludedMiddleasserts"pornotp"asalogicalstatement.The"or"inthis10.AllwomenarepoordriversorIammistaken.Iamnotmistaken.instanceisusedinthelimitedorexclusivesense.Forexample,"Anumber11.Anyonehandlingatoadwillgetwartsonhishand.Ihandledatoadiseitheranoddnumberoritisnotanoddnumber."Anotherexample,today."Silverisheavierthangoldorsilverisnotheavierthangold.Itcannotbe12.Allgoonsareloons.Thisisaloon.both."13.JoneslivesinDallasorhelivesinHouston.JonesdoesnotliveinDallas.Thesymbolforthe"exclusiveor"is"y.."Thetruthtableforthe"exclu-14.Allsquaresarerectangles.Thisisnotarectangle.siveor"follows.15.Ifa=b,thenac=bc.ac=Pbc.pqPIf.q16.IfREST,+-+thenRESl.~R~ST.0-017.IfBEXC,thenBEAC.BEAC.TTFTFTEachofthefollowinggivesthepatternforarrivingataconclusion.WriteFTTthestatementswhichcompletethepattern.FFF(I)IfBEAC,++thenBEAC.0--0(2)18.(3)ThenB~AG.(I)Ifx=4,thenyTheRuleforDenyingtheAlternativeisexpressedschematicallyby:Jq..=4.(2)x=4.(3)Then1.Porq1.Porq(1)Ifx=y,thenx=Pz.(2)x=y.20.2.not-q2.not-p(3)Then3.:.p3.:.q(I)21.(2)Ifa=Pb,thenac.(3)Thena=c.,jAsanexample,ifweacceptthestatements(a)"Thenumberkisoddorthe(I)Thisisanacuteoranobtusetriangle.(2)22.numberkiseven,"and(b)"Thenumberkisnoteven,"wemustthenconclude(3)Thenthisisanobtusetriangle.that(c)"Thenumberkisodd."(I)SEiffWewillusethesetwoprinciplesindevelopingproofsfortheoremslaterin23.orSElIT.(2)thisbook.(3)ThenSEfIT(I)24.(2)Iisparalleltom.(3)ThenInm=.0.Exercises25.Qllisnotparalleltom.(2)Inthefollowingexercisessupplyavalidconclusion,ifonecanbesupplied(3)Inm=P,0.bythemethodofmodusponensormodustollens.Assumethe"or"inthe26.QLAIr-LBeormLABC=P90.(2)AiJisnot-LBe.followingexercisestobetheexclusiveor.(Note.Youarenotaskedtodeter-.1(3)Thenminewhetherthepremisesorconclusionsaretrue.) 64FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGIC62.10.Converseofanimplication.Manystatementscanbeexpressedin]7.IfamanlivesinLosAngeles,helivesinCalifornia.converseform.Thisisdonebyinterchangingthe"if"andthe"then"18.Parallellinesinaplanedonotintersect.ofthestatement.19.Ifx=5,thenx2=25.Definition:Theconverseofp~qisq~p.20.IfBisbetweenAandC,thenmAC=mAB+mBG.Frequentlywearepronetoacceptastatementand,thenwithoutrealizing2.11.Logicalequivalence.Wehaveseenthattheconverseofatrueimplicait,infertheconverseofthestatement.Theconverseofastatementdoesnottiondoesnothavetohavethesametruthvalueasthatofthestatementbutalwayshavethesametruthvalueasthestatement.Anobviousexampleisofcourse,itmay.Iftwostatementsmutuallyimplyeachother,theyanthetruestatement"Allhorsesareanimals,"andthefalseconverse"Allsaidtobelogicallyequivalent.Logicallyequivalentstatementspresenanimalsarehorses."Brokenintoparts,the"if'ofthestatementis,"Thisisthesameinformation.ahorse,"whereastheconclusionis,"Thisisananimal."Theconverseofthestatement"AllHuftonsaregoodradios"is"IfaradioDefinition:Thestatementspandqareequivalentifpandqhavethesamtisagoodone,itisaHufton."Ingeometry,theconverseofthestatementtruthvaluesandmaybesubstitutedforeachother."Perpendicularlinesformrightangles"is"Iflinesformrightangles,theyIfpandqareequivalentstatements,weindicatethisbywritingp~q.areperpendicular."Inthiscase,boththestatementanditsconverseareThismeansp~qandq~p.Thetruthtableforequivalencecanbetrue.However,notethefollowingsyllogism.developedasfollows:1.P~'"~..Jj~'vPqP~qq~PP-q~.:.pTTTTTTFFTExercisesFYTTFFInthefollowingexercisesdetermine,ifpossible,thetruthorfalsityoftheFFTTTgivenstatement.Thenwritetheconverseofeachstatementanddetermine(ifpossible)thetrlltI1orfalsityyftht:«)111~I'S~.--.---------Thefollowingareequivalentstatements.I.Carrotsarevegetables.2.EveryU.S.citizenover21yearsofagehastherighttovote.3.Fordsarecars.p:LineIisparalleltolinem.4.Half-linesarerays.q:LinemisparalleltolineI.5.Nojournalistsarepoorspellers.6.Iftwoanglesareeacharightangle,theyarecongruent.Logicallyequivalentsentencesareoftenputintheform"ifandonlyif."7.Onlyamoronwouldacceptyouroffer.Thuswehave,8.Onlyparallellinesdonotmeet."IisparalleltomifandonlyifmisparalleltoI."Anotherobviousequivalenceisthedoublenegation,sinceadoublenega-9.Tosucceedinschoolonemuststudy.tionisequivalenttothecorrespondingpositivestatement.Thus,forevery]O.Onlyperpendicularlinesformrightangles.statementp,wehave1].Diamondsarehard.12.Ageometricfigureisasetofpoints.[not(not-f)]~p.13.Anequilateraltrianglehasthreecongruentsides.]4.Ifaislessthanb,thenbislargerthana.]Asanexample,ifpmeans"Threeisaprimenumber,"thenthedouble5.Ifx-y=1,thenxislargerthany.]negationofpisstated"Itisfalsethatthreeisnotaprimenumber."The6.Equilateraltrianglesareisosceles.twostatementsareequivalent. 66FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGIC67Exercises19.p:IftodayisSaturday,thentomorrowisSunday.Inthefollowingexercisesdeterminewhichpairsareequivalent.Notethat,!q:TomorrowisnotS~nday;~encetodayisnotSaturday.IinsomeexercisesjJandqaresimplestatements;inothers,pandqareimplica-I20.p:Ifa<~,ther~:1-bISnegatIve.tions.jq:Iffl-bISpOSJtIV~,thena>b.21.p:landmaretwolmesandlnm=cpoI.p:5isgreaterthan3.q:LinesIandmareparalleltoeachother.q:3islessthan5.22.p:Ifr,thennot-5.2.p:a+2b=4.q:Ifs,thennot-r.q:2a+4b=8.23.p:Ifnot-r,thens.3.p:Linelisperpendiculartolinem.q:Ifnot-s,thenr.q:Linemisperpendiculartolinel.24.p:Thefigureisatriangle.4.p:Lineslandmarenotparallel.q:Thefigureisthatformedbytheunionofthreelinesegments.q:Lineslandmintersect.2.12.Fourrulesofcontraposition.Logicallyequivalentstatementsmay5.p:Ifitisadog,ithasfourlegs.besubstitutedforeachotherwhenevertheyoccurinadiscourse.Oneq:Ifitdoesnothavefourlegs,itisnotadog.particulartypeofequivalencehasgreatvalueinthestudyoflogic,namely,6.p:Perpendicularlinesformrightangles.contraposition.q:Rightanglesformperpendicularlines.7.p:Adiameterisachord.Definition:Thestatementnot-q~not-piscalledthecontrapositiveoftheq:Achordisadiameter.statementofp~q.8.p:x=y.Therearefourcommontypesofcontraposition.Astudyofthefollowingq:y=x.fourequivalenceswjUrevealthatthecontrapositiveisthenegationofthe9.p:Fornumbersa,b,c,a=b.clausesoftheconverse,aswellastheconverseofthenegationoftheclausesq:Fornumbersa,b,c,a+c=b+c.oftheoriginalimplication.10.1):Thepresentwasexpensive.q:Itisnottruethatthepresentwasexpensive.II.p:IfheisanativeofSpain,heisanativeofEurope.q:IfheisnotanativeofEurope,heisnotanativeofSpain.cIfnot-p,thennot-q12.p:Iftwolinesmeettoformrightangles,theyareperpendicular.~.,;(not-p~not-q)~(q~P).Ifq,thenpq:Iftwolinesarenotperpendicular,theydonotmeettoformrigh~angles.3.!.!),thennot-q13.p:PointsRandSareonoppositesidesoflinel.Ifq,thennot-p;(jJ~not-q)~(q~not-p).!q:LinesegmentRSintersectslinel.Ifnot-p,thenq14.jJ:BisbetweenAandC.4.q:BEAC,BA,BC.Ifnot-q,thenp;(not-p~q)~(not-q~p).""""IS.p:landmaretwolinesandAElnm.Thestudentshouldstudythefourtypesuntilheissatisfiedthatifyouq:LinelandlinemintersectatpointA.accepteitheroneofapairofcontrapositivesastrue,youmustacceptthe16.p:R~Sf.+-+i~therastruealso.Thefollowingexamplesillustratetheapplicationsoftheq:RliesononesideofST.OUrtypes.17.p:LRSTisanacuteangleandLABCisanobtuseangle.I.Ifhecanvote,thenheisover21q:mLABC>mLRST..IfheISnotover21yearsofage,thenhecannotvote.Yearsofage.18.p:Verticalanglesarecongruent.2I.flandmarenotperpendicular,theydonotintersectatrightangles.q:Iftheanglesarenotverticalangles,thentheyarenotcongruent.~-~---- 68FUNDAMENTALSOFCOLLEGEGEOMETRYELEMENTARYLOGIC6~Iflandmintersectatrightangles,theyareperpendicular.Ifx=)',thenx2=)'23.Ifhedrives,heshouldnotdrink.17.Ifx2=y2,thenx=y'Ifhedrinks,heshouldnotdrive.4.Ifthenaturalnumberisnoteven,itisodd.If!donotstudy,Iwillnotpassthiscourse.18.Ifthenaturalnumberisnotodd,itiseven.If!study,Iwillpassthiscourse.Ifthisisrhombus,itisnotatrapezoid.Theequivalenceofcontrapositivestatementsisshownbythefollowing19.truthtable.ThenumbersundereachcolumnindicatestheorderofeachIfthisisatrapezoid,itisnotarhombus.step.Ifo'/'Ii,thenc#d;c#d20.a#b(p~q)-(~q~~p)Ife'/'d,thena#b;c#d21.TTTTFTFa'/'bTFFTTFFIfe~AB,thenC~A11FTTTFTT22.<>--<>.IfeEAB,thenCEABFTFTTTTIfIisnot11m,thenInm=apoint23....314232IfInmISnotapomt,tenhIIImIfIII..m,In11/=.0Exercises24..If!n1I/=.0,lllmEachexercisecontainsaconditionalstatement.Form(a)itsHitisThanksgivingDay,themonthisNovember.(b)itscontrapositive,and(c)theconverseofitscontrapositive.2::>."ItisnotDecemberItisThanksgivi"ugDay.I.1fTERX,thenTERX.---+~2.1fTERX,thenTERX.3.If~GEA&itflet1£'-Eu"1B.--4.Ifa+c=b+c,thena=b.5.Ifa+b=0,thena=-b.6.Ifa+b=c,thencisgreaterthana.7.IwillpassthiscourseifIstudy.8.Ifheisanalien,heisnotacitizen.9.Parallellineswillnotmeet.10.IfthisisnotaZap,itisaZop.II.Ifthefigureisnotarectangle,itisnotasquare.12.IfheisnotaEuropean,heisnotanativeItalian.13.Ifthetriangleisequilateral,itisequiangular.14.Goodcitizensdonotcreatedisturbances.Inthefollowingexercisesdeterminewhichoftheconclusionsarevalid.Goodcitizensdonotcreatedisturbances.Idonotcreatedisturbances.15.Iamagoodcitizen.IfIstudy,Iwillpassthiscourse.Istudy.16.Iwillpassthiscourse. converseofatruestatementisalwaystrue.0Thenegationofafalsesta.tementmayresultinatruestatement.~l:The22."Januaryhas32daysor4ISlessthan5"isatruestatement.23."not-notp"hast~esamemeaningas"p."24.IfPistrue,not-pISalsotrue.negationof"NoAisB"is"EveryAisB."25.Thenegationof"EveryLakisaLuk"is"NoteveryLukisaLak."26.The27.(P~Q)~(Q~P)028.not(Porg)~(notpornotg).29.not(Pandg)~(notpandnotg).30.(P~g)~(notp~notg).SummaryTest31.(notp~notg)~(P~g).32.(P~notg)~(g~notP).33.(notp~g)~(notg~P).34.jJ~if35.P~q36.P~qqpnotgIneachofthefollowingindicatewhetherthestatementisalwaystrue(mar:.p:.q:.notpT)ornotalwaystrue(markF).37.P~q38.p~q39.jJ~q1.Validconclusionscanresultfromfalse(untrue)basicassumptions.notp~~2.Theconverseof"IntriangleRST,ifm(RT)>m(RS),thenmLS>mLT:.p:.notq:.ifis"IntriangleRST,ifmLS>mLT,thenm(RS)>m(RT).",40.p~q41.p~qorq~p42.notp~q3.Theconverseof"Ifyoueattoadstools,you'llgetsick"is"Youwillg~:.p~qandq~p:.p~q:onotq~psickifyoueattoadstools."4."Closethedoor!"isastatement.5."ItiscoldandIamfreezing"isastatement.6.Givenpistrue,gisfalse.Then"pandg"isfalse.7.Givenpisfalse,gistrue.Then"pandg"isfalse.8.Givenpistrue,gisfalse.Then"porg"isfalse.9.Givenpisfalse,gistrue.Then"porg"isfalse.10."porg"iscalledaconjunctionofpandg.I1.IfPisfalse,thennot-pistrue.'.'12.Anegationofthestatement"Noteverystudentissmart"is"Noteve~~studentisstupid."13.Anegationofthestatement"aequals2andbequals3"is"adoesnoteq2andbdoesnotequal3."14.Anegationof"Someblindmencansee"is"Atleastoneblindman«see."15.Thenegation"not(porg)"hasthesamemeaningas"notpornotgo"16."Not(pandg)"meansthesameas"not(porg)."17."Notpornotg"meansthesameas"not(pandg)."18.Ifanimplicationistrue,itsconverseisalsotrue.19.Theconverseof"Ifant=cp,thenat,thenant~cpo".,IIt"is"IfaII7170 Definition:Arealnumberispositiveiffitisgreaterthanzero;itisnegativeilfitislessthanzero.31Wesaythata>biffa-bisapositivenumber.Similarly,a"andfor"isnotlessthan"is"b.0-2(additionproperty).(a(c-b).0-4(multiPlicationproperty).(a0)~acbc.0-5(divisionproperty).(a0)~alcclb;3.1.Propertiesofrealnumbers.Inyourfirstcourseinalgebra,youj(ablc1cia0)~c>a.presentrealnumbers.Hereafter,youcanrefertothesepropertieseitherbvnameorbyrepeatingthepropertywhenaskedtosupportdeductions.~PropertiesofafieldmadeaboutF€-al-numbers--------------------ThefollowingadditionalpropertiesoftherealnumbersystemarecalledEqualityProperties"fieldproperties."E-I(reflexiveproperty).a=a.OperationsofAdditionE-2(symmetricproperty).a=b~b=a.F-l(closureproperty).a+bisauniquerealnumber.E-3(transitiveproperty).(a=b)1(b=c)~a=c.F-2(associativeproperty).(a+b)+c=a+(b+c).E-4(additionproperty).(a=b)1(c=d)~(a+c)=(b+d).F-3(commutativeproperty).a+b=b+a.E-5(subtractionproperty).(a=b)1(c=d)~(a-c)=(b-d).F-4(additivepropertyofzero).Thereisauniquerealnumber0,theadditiveE-6(multiplicationproperty).(a=b)1(c=d)~ac=bd.identityelement,suchthata+0=0+a=a.F-5(additiveinverseproperty).Foreveryrealnumbera,thereexistsarealE-7(divisionproperty).(a=b)1(c=d#0)~~=!l.number(-a),theadditiveinverseofa,suchE-8(substitutionproperty).AnyexpressionmaybereplacedbyaneqUivalen..thata+(-a)=(-a)+{1=O.expressioninanequationwithoutchangingthetruthvalueoftheequation..,,<.',OperationsofMultiplicationThesymbolfor"isgreaterthan"is">"andfor"islessthan"isThus,a>bisread"aisgreaterthanb."Itshouldbenotedthata>ban.~.F-6(closureproperty).a'bisauniquerealnumber.b3x+9.following?2.3x-15=4x-8.2.5x>3x+16.1.4+3=3+4.2.5+(-5)=O.3.3x=4x+7.3.2x>16.3.6+0=6.4.7'I=7.4.-x=7.4.x>8.5.2(5+4)=2.5+2.4.6.5.2=2.5.5.x=-7.II29.1.3x-9<7x+15.30.1.2(x-3)>5(x+7).7-24.Namethepropertyoftherealnumbersystemwhichwillsupportthe2.3x<7x+24.2.2x-6>5x+35.indicatedconclusion.3.-4x<24.3.2x>5x+41.7.lfx-2=5,thenx=7.4.x>-64.-3x>41.8.lf3x=12,thenx=4.5.x<-'Y.9.lf7=5-x,then5-x=7.10.lfa+3=7,thena=4.3.2.Initialpostulates.Inthiscourse,weareinterestedindeterminingandprayinggeometricfacts.Wehave,withtheaidoftheundefinedgeometricII.1[2a+5=9,then2a=4.12.lfa+b=10,andb=3,thena+3=10.concepts,definedasclearlyandasexactlyaswecouldothergeometnccon-ceptsandterms.Wewillnextagreeonorassumecertainpropertiesthatcan13.Ifix=7,thenx=14.beassignedtothesegeometricfigures.Theseagreed-uponpropertieswe14.5,(t)=1.willcallpostulates.Theyshouldseemalmostobvious,eventhoughtheymay15.Ifa+3<8,thena<5.bedifficult,ifnotimpossible,toprove.Thepostulatesarenotmadeupat16.Ifx=yandy=6,thenx=6.random,buthavebeencarefullychosentodevelopthegeometryweintendto17.Ifx>yandz>x,thenz>y.deyelop.Withdefinitions,propertiesoftherealnumbersystem,andpostu-18.Ifa-2>10,thena>12.19.If-3x<15,thenx>-5.latesasafoundation,wewillestablishmanynewgeometricfactsbygiving20.1/2+V4isarealnumber.logicalproofs.Whenstatementsaretobelogicallyproved,wewillcallthemtheorems.21.(S.t)'12=5.(t'12).Onceatheoremhasbeenproved,itcanbeusedwithdefinitionsandpostu-22.(17+18)+12=17+(18+12).latesinprovingothertheorems.23.Iftx>-4,thenx>-12.Itshouldbeclearthatthetheoremswhichwecanprovewill,toagreat24.3(y+S)=3y+IS.extent,dependuponthepostulatesweagreetoenumerate.Alteringtwo25-30.NamethepropertyofrealnumberswhichjustifyeachofthenumberedOrstepsinthefollowingproblems.~hreepostulatescancompletelychangethetheoremsthatcanbeprovedinagIvengeometrycourse.Hence,weshouldrecognizetheimportanceoftheIllustrativeProblem.8-3x=2(x+6).selectionofpostulatestobeused.Solution DEDUCTIVEREASONING7776FUNDAMENTALSOFCOLLEGEGEOMETRYTheorem3.1Thepostulateswewillagreeonwillingreatpartreflecttheworldaboutus.3.3.Iftwodistinctlinesinaplaneintersect,thentheirintersectionisatDefinition:Astatementthatisacceptedasbeingtruewithoutproofismost,onepoint.calledapostulate.Supportingargument.LetIandmbetwodistinctlinesthatintersectatS.postulate1.Alinecontainsatleasttwopoints;aplanecontainsatleastthreeCsingthelawoftheexcludedmiddle,weknowthateitherlinesIandminter-pointsnotallcollinear;andspacecontainsatleastfourpointsnotallcoplanar.sectinmorethanonepointortheydonotPostulate2.Foreverytwodistinctpoints,thereisexactlyonelinethatcontainsbothintersectinmorethanonepoint.Iftheypoints.intersectinmorethanonepoint,suchasat~RandS,thenlineIandlinemmustbetheNoticethatthispostulatestatestwothings,sometimescalledexistenceandsameline(applyingPostulate2).Thisunzqueness:~contradictsthegivenconditionsthatIandmoR1.Thereexistsonelinethatcontainsthetwogivenpoints.aredistinctlines.Therefore,applyingtheTheorem3.1.2.Thislineisunique;thatis,itistheonlyonethatcontainsthetwopoints.rulefordenyingthealternative,linesIandmintersectin,atmost,onepoint.Postulate3.Foreverythreedistinctnoncollinearpoints,thereisexactlyoneplanethatcontainsthethreepoints.Theorem3.2Postulate4.Ifaplanecontainstwopointsofastraightline,thenallpointsofthe3.4.IfapointPliesoutsidealineI,exactlyoneplanecontainsthelineandlinearepointsoftheplane.thepoint.Postulate5.Iftwodistinctplanesintersect,theirintersectionisoneandonlyoneSupportingargument.ByPostulateI,lineIcontainsatleasttwodifferentline(seeFig.3.1).points,sayAandB.SincePisapointnotonI,wehavethreedistinctnon-collinearpointsA,B,andP.Postulate3,then,assurestheexistenceanduniquenessofaplaneMthroughlineIandpointP.p.~Fig.3.1.Theorem3.2.Theorem3.33.5.Iftwodistinctlinesintersect,exactlyoneplanecontainsbothlines.Withtheabovepostulateswecanstartprovingsometheorems.ThesefirstSupportingargument.LetQbethepointwherelinesIandmintersect.theoremswillstatewhattomostofuswillseemintuitivelyobvious.Un-Postulate1guaranteesthatalinemustcontainatleasttwopoints;hence,fortunately,theirformalproofsgettrickyandnottoomeaningfultothetheremustbeanotherpointonIandanotherpointonm.Letthesepointsgeometrystudentbeginningthestudyofproofs.Consequently,wewillbeletteredRandP,respectively.Postulate3tellsusthatthereisexactlygiveinformalproofsofthetheorems.Youwillnotberequiredtoreproduce'oneplanethatcontainspointsQ,R,andP.WealsoknowthatbothIandmthem.However,youshouldunderstandclearlythestatementsofthe)jll1ustlieinthisplanebypostulate4.theorems,sinceyouwillbeusingthemlaterinprovingothertheorems. DEDUCTIVEREASONING7978FUNDAMENTALSOFCOLLEGEGEOMETRY16.Explainhow,withastraightedge,itispossibletodeterminewhetherallpointsofthetORofatablelieinoneplane.17.IfinplaneMN,Xi3.1line~,Ac.1linem,andAisonm,doesitneces-+--+~sarilyfollowthatAB=Ae?18.Isitpossiblefortheintersectionoftwoplanestobealinesegment?Explainyouranswer.19.Usingtheaccompanyingdiagram(a3-dimensionalfigure),indicatewhichsetsofpointsare(1)collinear,(2)coplanarbutnotcollinear,(3)notTheorem].3.coplanar.F(a){A,C,D}Summarizing.Aplaneisdeterminedby1.Threenoncollinearpoints.(b){D,A,F}2.Astraightlineandapointnotontheline.3.Twointersectingstraightlines.(c){F,G,A}Exercises(d){F,D,G}1.Howmanyplanescanbepassed(a)throughtwopoints?throughthree(e){F,B,C,E}Bcpointsnotinastraightline?Ex.19.2.Whatfigureisformedattheintersectionofthefrontwallandthefloorofaclassroom?3.Holdapencilsothatitwillcastashadowonapieceofpaper.WillthelWhichofthefollowingchoicescorrectlycompletesthestatement:shadowbeparalleltothepencil?Threedistinctplanescannothaveincommon(a)exactlyonepoint,(b)4.Howmanyplanes,ingeneral,cancontainagivenstraightlineandapoint!exactlytwopoints,(c)exactlyoneline,(d)morethantwopoints.notontheline?'J.Additionalpostulates.IuChapter1wedi!>cu!>sedtherealnumberline.5.Howmanyplanescancontainagivenstraightlineandapointnotontheline?]eshowedthecorrespondencebetweenpointsonthenumberlineandthe6.Whyisatripod(threelegs)usedformountingcamerasandsurveying11numbers.Inorderthatwemayuseinsubsequentdeductiveproofstheinstruments?conclusionswearrivedat,wewillnowrestatethemaspostulates.7.Howmanyplanesarefixedbyfourpointsnotalllyinginthesameplane?Postulate6.(therulerpostulate).Thepointsonalinecanbeplacedinaone-to-8.Whywillafour-leggedtablesometimesrockwhenplacedonalevelfloor?onecorrespondencewithrealnumbersinsuchawaythat:9.TwopointsAandBlieinplaneRS.WhatcanbesaidaboutlineAB?I.Foreverypointofthelinetherecorrespondsexactlyonerealnumber;10.Iftwopointsofastraightrulertouchaplanesurface,howmanyother1!2.foreveryrealnumber,therecorrespondsexactlyonepointoftheline;andpointsoftherulertouchthesurface?3.thedistancebetweentwopointsonalineistheabsolutevalueofthedifference11.Canastraightlinebeperpendiculartoalineinaplanewithoutbeing!betweenthecorrespondingnumbers.perpendiculartotheplane?12.Cantwostraightlinesinspacenotbeparallelandyetnotmeet?Explain.Postulate7.Toeachpairofdistinctpointstherecorrespondsauniquepositive13.OnapieceofpaperdrawalineAB.PlaceapointPonAB.Inhownumber,whichiscalledthedistancebetweenthepoints.manypositionscanyouholdapencilandmakethepencilappearper-ThecorrespondencebetweenpointsonalineandrealnumbersiscalledpendiculartoABatP?~hecoordinatesystemfortheline.Thenumbercorrespondingtoagivenpoint14.Arealltrianglesplanefigures?Givereasonsforyouranswer.IScalledthecoordinateofthepoint.InFig.3.2,thecoordinateofAis-4,15.J:I°w~nLd~rent~nesaredeterminedbypairsofthefourdifferentofBis-3,ofCis0,ofEis2,andsoon.bnesAP,BP,CP,andDPnothreeofwhicharecoplanar?--- 80FUNDAMENTALSOFCOLLEGEGEOMETRYDEDUCTIVEREASONING81ABCDEFas"themeasureofthewholeisequaltothesumofthemeasuresofitsparts."..I...~L~-6-5-4-3-2-10123456postulate15.Asegmenthasoneandonlyonemidpoint.Fig.3.2.postulate16.Ananglehasoneandonlyonebisector.ExercisesPostulate8.Foreverythreecollinearpoints,oneandonlyoneisbetweentheothertwo.Thatis,ifA,B,andCare(distinct)collinearpoints,thenoneandonly1.Whatpointhasacoordinateof-3?oneofthefollowingstatementsistrue:(a)AliesbetweenBandC;(b)Blies2.WhatisthedistancefromBtoE?betweenAandC;(c)CliesbetweenAandB.3.WhatisthemBD?Postulate9.IfAandBaretwodistinctpoints,thenthereisatleastoneCpoint4.WhatisthemDA?suchthatCEAB.Thisis,ineffect,sayingthateverylinesegmenthasat5.WhatisthecoordinateofthemidpointofBE?leastthreepoints.ARCDEFIjPostulate10.IfAandBaretwodistinctpoints,thereisatleastonepointDsuch-6-4-20246thatABCAD.Postulaten.ForeveryAlJandeverypositivenumbernthereisoneandonlyaniExs.l-IO.pointPofABsuchthatmAP=n.Thisiscalledthepointplottingpostulate.6.WhatisthecoordinateoftheendpointofFA?Postulate12.IfABisarayontheedgeofthehalf-planeh,thenforeverynbe-7.WhatisthemBC+mCD+mDE?DoesthisequalthemBE?8.WhatisthemBA?tween0and180thereisexactlyonerayAP,withPinh,suchthatmLPAB=n.9.IsthecoordinateofpointAgreaterthanthecoordinateofpointD?Thisiscalledtheangleconstructionpostulate.10.DoesmBD=mDB?Postulate13.(segmentadditionpostulate).Asetofpointslyingbetweenthe11.a,b,c,arethecoordinatesofthecorrespondingpointsA,B,C.Ifa>cendpointsofalinesegmentdividesthesegmentintoasetofconsecutivesegmentstheandc>b,whichpointliesbetweentheothertwo?sumofwhoselengthsequalsthelengthofthegivensegment.12.IfTisapointonRS,completethefollowing:Thus.inFig.3.3,ifA,B,C,Darccollincar,thenmAB+mBC+/fICD=(3)mlIT+mTS=(b)mRS-mTS=..r:r.-A,B,andCarethreecollinearpoints,mBC=15,mAB=11.WhichpointABCDcannotliebetweentheothertwo?Fig.3.3.14.R,S,Tarethreecollinearpoints.IfmRSmLEBG.4.LEBCisthecomplement4.Given.Prove:mLABE>mLDBC.ofLEBA.Proof5.mLEBC+mLEBA=90.5.Twoanglesarecomplementaryiffthesumoftheirmeasuresis90.6.mLEBC+mLEBA=mLABC.6.Angleadditionpostulate.IllustrativeExamjJle2.7.mLABC=90.7.Substitutionpropertyofequality(orTheorem3.5).STATEMENTSREASONSLABC=-mLAEB.8.(a=c)/(b=c)~a=b.9.LABC==LAEB.9.Angleswiththesamemeasure1.BEandBJjareraysdrawnfrom1.Given.arecongruent.fromthevertexofLABCasshown.A,B,C,D,Eareco-planar.Exercises2.mLABD>mLEBC.2.Given.InthefOllowingexercisescompletetheproofs,usingforreasonsonlythe.3.mLABD+mLDBE>mLEBC+3.Additivepropertyoforder.gIVen,definitions,propertiesoftherealnumbersystem,postulates,theorems,mLDBE.andcorollarieswehaveprovedthusfar.4.mLABD+mLDBE=mLABE;4.Angleadditionpostulate.1.ProvemLEBC+mLDBE=mLDBC.Theorem39..2.ProvethecorollarytoTheorem3.10.5.mLABE>mLDBC.5.Substitutionpropertyoforder.3.Given:CollinearpointsA,B,C,Dasshown;mAC=mBD.Prove:mAB=mCD.3.19.IllustrativeExample3:Given:BJj..lAG.LEBCisthecomplementofLEBA.--ULL4Prove:LABCABcD==LAEB.ProofExs.3-4. 88FUNDAMENTALSOFCOLLEGEGEOMETRYDEDUCTIVEREASONING894.Given:CollinearpointsA,B,C,Dasshown;mAC>mBD.CProve:mAB>mCD.10.Given:PointsDandElieonsidesACand5.Given:LABCwithjjJjandliEasshown.BCofLABCasshown;mAD=A,B,C,D,Earecoplanar;mBE;mDC=mEe.mLABD=mLEBe.prove:LABCisisosceles.Prove:mLABE=mLDBC.6.Given:LABCwithBl5andBEasshown.A,B,C,D,Earecoplanar;ABmLABE=mLDBC.Ex.10.Prove:mLABD=mLCBE.Exs.5-6.EI].Given:A,B,C,Darecollinearpoints;7.Given:mPS=mPR;mTS=mQR.mLEBC=mLECB.Prove:mPT=mPQ.Prove:IIlLABE=mLDCE.ELABcD""pAJEX.n.QEx.7.c/m~__-PointsDandEonsidesACanclBC8.Given:mLABC=mLRSTandofLABCasshown;mLBAC=mLABD=mLRSP.R~mLABC;mLCAE=mLCBD.Prove:mLDBC=mLPST.Prove:IIlLEAB=mLDBA.AEx.8.AEX./2.9.Given:LABC~LRST;Lcf>~L8;La~L{3.Prove:LABD~LRSP.~/D~/T13.Given:Be..lAB;LBACandLCADarecomplementary.DlLJC~~Prove:mLDAB=mLABe.BARSABEx.9.Ex.13.----------------------------------------------------------------- DEDUCTIVEREASONING9190FUNDAMENTALSOFCOLLEGEGEOMETRYDTheorem3.12Verticalanglesarecongruent.3.21.14.Given:AD1-AB;BC1-AB;LB==LC.ci1!en:ABandCDarestraightlinescBintersectingatE,formingProve:LA==LC.verticalanglesLxandLy.Ly.AConclusion:Lx==EX./4.AProofTheorem3/2STATEMENTSREASONS15.Given:BC1-AB;LCisthecomplementofLABD.Prove:mLC=mLDBC.1.ABandCDarestraightlines.1.Given.2.LCEDandLAEBarestraight2.Definitionofstraightangle.Aangles..Ex./5.3.LxandLraresupplementary3.Theorem3.11.angles.Theorem3.114.LyandLraresupplementary4.Theorem3.11.angles.3.20.Twoadjacentangleswhosenoncommonsidesformastraight5.Lx==Ly.5.Supplementsofthesameanglearesupplementary.arecongruent.Given:LABDandLDBCareadjacentangles.LABCisastraightangle.cConclusion:LABDisasupplementofLDBC.Proof:erpendicularlinesformfourrightangles:-.AGiven:CD1-ABatO.A0BTheorem3.1/.Conclusion:LAOC,LBOC,LEOD,andLAODarerightangles.STATEMENTSREASONSProof1.LABDandLDBCareadjacent1.Given.angles.D2.LABCisastraightangle.2.Given.3.mLABC=180.3.ThemeasureofastraightaTheorem3./3.S'fA'fEMENTSis180.----REASONS4.mLABD+mLDBC=mLABC.4.AngleadditionpostUlate.1.CD5.a=b1b=c1.AiJatO.5.mLABD+mLDBC=180.~a=c.2.L..BOC1.Given.6.Ifthesumofthemeasureso~isarightangle.2.1-linesformarightangle.6.LABDisasupplementofLDBC.3.'fIlL..BOCanglesis180,theangleS.,=90.3.Themeasureofarightangleissupplementary.90.-------------------------------------------------------------------..- 92FUNDAMENTALSOFCOLLEGEGEOMETRYDEDUCTIVEREASONING934.LAOBisastraightangle.4.Definitionofastraightangle.5.LBOCandLAOCareadjacent5.Definitionofadjacentangles.angles.Ac6.LBOCandLAOCaresupple-6.Theorem3.11.mentary.7.mLBOC+mLAOC=180.7.Thesumofthemeasuresoftw..supplementaryanglesis180.DF8.mLAOC=90.8.Subtractionpropertyofequalit:9.mLAOD=mLBOC;9.Verticalanglesarecongruent.mLBOD=mLAOC.10.mLAOD=90;mLBOD=90.10.Substitutionpropertyofequalit'l11.LAOC,LBOC,LBOD,LAOD11.Statements2,8,9,anddefiIllustrativeExam/)le.arerightangles.tionofrightangle.3.24.IllustrativeExample:cTheorem3.14.Given:AC,DFandGHarestraightlines.LGBC==LBEF.Prove:LABG==LDEB.3.23.Iftwolinesmeettoformcongruentadjacentangles,theyareperpendicular.Proof"+---i-Given:CDandABintersectat0;STATEMENTSREASONSALAOC==LBOC.0Prove:CD1-}[jJ.1.ACisastraightline.I.Given.Proof2.LABCisastraightangle.2.Definitionofstraightangle.3.LABGisthesupplementofLGBC.3.Theorem3.11.1--:-DFisa-+-'-Theorem3.14.5.LDEFisastraightangle.5.Sameasreason2.STATEMENTSREASONSD6.LDEBisthesupplementofLBEF.6.Theorem3.11.7.LGBC7.Given.==LBEF.1.LAOC==LBOC.1.Given.8.LABG==LDEB.8.Supplementsofcongruentangles2.mLAOC=mLBOC.2.Congruentarecongruent.sures.3.mLAOC+mLBOC=mLAOB.3.Angleadditionpostulate.4.LAOBisastraightangle.4.Definitionofstraightangle.Exercises5.mLAOB=180.5.Themeasureofastraightaniis180.Inthefollowingexercisesgiveformalproofs,usingforreasonsonlythe.6.mLAOC+mLBOC=180.6.a=b1b=c~a=c.gIvenstatements,definitions,postulates,theorems,andcorollaries.7.mLBOC+mLBOC=180.7.Substitutionpropertyofequali']1.Given:CDandCEarestraightlines.LCAB8.mLBOC=90.8.Divisionpropertyofequality.~==LCBA.Prove:LBAD9.LBOCisarightangle.9.Definitionofrightangle..==LABE.~~2.Given:AG,DE,and!iFasshowninthefigure.LABEisthesupplement10.CD1-AB.10.DefinitionofperpendicuJofLBCF.lines.Prove:LDCB==LABE. DEDUCTIVEREASONING9594FUNDAMENTALSOFCOLLEGEGEOMETRYc6.Given:JiB,t:JJ,andUareDstraightlines;La==LbE~O/!Prove:EFbisectsLBOD.<:.-A/D"'AEx.6.CED7.Given:La==Lb;Ex.i.Ex.2.Le==Ld;LA==LB.hAdProve:La==Le.~A.BAEx.7.3.Given:Aif,CD,andEftareDcstraightlines;Lb==Le.c8.Given:AD1-AB;BC1-AB;Prove:La==Le.LBAC==LABD.Prove:LDAC==LCBD.EX.3.ABEx.8.9.Given:LABC;CDbisectsLACB;LAisthecomplementofLACD;4.Given:AB,EC,andCDarestraightlines;La==Ld.LBisthecomplementofLBCD.Prove:LA==LB.Prove:Lb==Le.EX.4.ADBEx.9.D5.Given:Ai!andBi5intersectingatC;10.Given:mAD=mBD;LAisthecomplementofLACB;AmAE=mED;LEisthecomplementofLDCE.mBC=mCD.Prove:LA==LE.Prove:mED=mCD.A~BEX.5.Ex10. 96FUNDAMENTALSOFCOLLEGEGEOMETRYDEDUCTIVEREASONING9711.Given:mAD=mBD;mAE=mBe.16.Given:AlJ..1CDat0;Prove:mED=mCD.DmLBOE=mLDOF.12.Given:mAE=mBC;Prove:OF..1DE.AE'jmED=mCD.ProofProve:mAD=mBD.¥13.Given:mED=rnCD;6Ex.16.mAE=rnED;STATEMENTSREASONSABmBC=rnCD.Exs.II-I3.Prove:mAE=mBe.1.AB..1CD.1.Why?2.LBODisarightL.2.Why?3.mLBOD=90.3.Why?4.rnLBOE+mLEOD=mLBOD.4.Why?D~5.mLBOE+mLEOD=90.5.Why?I6.mLBOE=mLDOF.14.Given:LABCisastraightL;/.6.Why?EBE7.mLDOF+mLEOD=90.bisectsLABD;/7.Why?~---FBbisectsLCBD./8.rnLDOF+mLEOD=mLEOF.8.Why?-"""""""""'~rl~IS9.mLEOF=90~c)Pm1.lR."-~~QJJF._---------------X'-Why-A10.:.OF..1DE.-----------------------------(Hint:rnLx+mLy+rnLr+rnLs=?;10.Why?Ex.14.IrnLx=rnL?;rnLr=rnL?).DE17.Given:AlJ..1CD;Ol..1OF.F15.Given:;m..1Cf5at0;Prove:LBOE~LFOD.mLBOE=rnLlJOF.ProofABProve:rnLEOD=rnLAOF.AProofEx.17.cCSTATEMENTSREASONSEx.15.STATEMENTSREASONS1.AB.l...1.Why?2.DE..1....2.Why?1.;m..1CD.1.Why?3.LBODisa...;LFOEisa....3.Why?2.LBODandLAODarerightAi.2.Why?4.mLBOD=...;mLFOE=....4.Why?3.rnLBOE=rnLDOF.3.Why?5.mL?+mL?=mLBOD.5.Why?4.mLBOD=rnLAOD.4.Why?6.mLBOE+mLDOE=90.6.Why?5.rnLEOD+rnLBOE=mLBOD.5.Why?7.mL?+mL?=mLFOE.7.Why?6.rnLAOF+mLDOF=rnLAOD.6.Why?8.mL?+mL?=90.8.Why?7.rnLEOD+mLBOE=rnLAOF7.Why?9.LBOEisthe...ofLDOE.9.Why?+mLDOF.10.LFODisthe...ofLDOE.10.Why?8.:.mLEOD=mLAOF.8.Why?11.LBOE~LFOD.11.Why?-------------------------------------- 20.Itisnotpossibleforverticalanglestobeadjacentangles.21.Itispossibleforthreelinestobemutuallyperpendicular.22.Aperpendicularisalinerunningupanddown.23.Iftwoanglesarecomplementary,theneachofthemisacute.24.Iftwoanglesaresupplementary,thenoneofthemisacuteandtheotherisobtuse.25.Twoanglesareverticalanglesiftheirunionistheunionoftwointer-sectinglines.Test2COMPLETIONSTATEMENTSSummaryTests1.Astatementconsideredtruewithoutproofiscalleda(n)2.Iftwoanglesareeithercomplementsorsupplementsofthesameangletheyare.3.Thesidesofarightanglearetoeachother.4.ThepairsofnonadjacentanglesformedwhentwolinesintersectareTest1called5.A(n)anglehasalargermeasurethanitssupplement.TRUE-FALSESTATEMENTS6.Thebisectorsoftwocomplementaryadjacentanglesformsanangle1.Oneplaneandonlyoneplanecancontainagivenlineandapointnotonwhosemeasureistheline.7.AngleAisthecomplementofananglewhosemeasureis42.AngleBis2.Thenumberiisarealnumberthatisnotrational.thesupplementofLA.ThenthemeasureofLBis3.Everyangleiscongruenttoitself.8.PointBliesonlineRS.LineABisperpendiculartolineRS.Then4.Twoacuteanglescannotbesupplementary.IIlLABR=5.Tile!t::;~exactlyoneplanecontainingagi~J.1woanglescomplementarytothesameangleare6.Thedistancebetweentwopointsisapositivenumber.10.Thedifferencebetweenthemeasuresofthesupplementandcomple-7.Apostulateisastatementthathasbeenproved.mentofanangleis8.Supplementaryanglesarecongruent.II.Thebisectorsofapairofverticalanglesformaangle.9.Thebisectorsoftwoadjacentsupplementaryanglesareperpendicular12.Themeasureofananglethatiscongruenttoitscomplementistoeachother.13.Themeasureofananglethathashalfthemeasureofitssupplementis10.Verticalangleshaveequalmeasures.11.Theabsolutevalueofeverynonzerorealnumberispositive.14.Thesumofthemeasuresoftwoadjacentanglesformedbytwointersect-12.Iftwolinesintersect,therearetwoandonlytwopointsthatarecontainedinglinesisbybothlines.15.Foreverythreedistinctnoncollinearpoints,thereisexactlyone13.Iftwolinesintersecttoformverticalanglesthataresupplementary,thethatcontainsthethreepoints.verticalanglesarerightangles.16.IfrnLAdonealJMlc;«;tJfm~~indcf-18.ifCA11-AB,thenLAMC:::::LBi+IC.19.IfCMbisectsLACB,thenAM==BM.20.IfFbisectsAE,thenFbisectsBD.CF21.IfLAFM==LBFM.ThenLCFA==LCFB.22.LAFCandLBFCareverticalangles.MA~BD~EExs.16-22.Fig.4.1.Exercises(B)Byusingthetheoremsoncongruence,whatconclusionscanbedrawnin]Thismatchingofverticesandsidesofageometricfigureiscalledaone-to-eachofthefollowingexercises?Writeyourconclusionsandreasonsinthdonecorrespondence.Thematchedpartsarecalledcorrespondingparts.samemannershowninthefollowingexample.EDThuswespeakofcorrespondingsidesandcorrespondingangles.Thematching---upschemeofcorrespondingverticescanbeshownbythesymbolism:A~D,IIIu,trativeexample,Given:LAEDLCDE;LDEBLEDB.YIB~E,C~F.WecanalsoshowthismatchingbywritingABC~DEF.====Conclusion:LAEBThusgivenacorrespondenceABC~DEFbetweentheverticesoftwo==LCDB.jRemon,An~le,nh"actiontheo,em.0.Dtriangles,ifeachpairofcorrespondingsidesarecongruent,andifeachpairIABofcorrespondinganglesarecongruent,thenthecorrespondenceABC~lllustrativeExample.DEFisacongruencebetweenthetwotriangles. 108FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES109TTwotrianglescanbematched-upsixways.Otherwaysofmatching6ABCone-tn-onecorrespondenceinwhicheveryvertexand6DEFare:ismatchedwithitselfiscalledtheidentitycongruence.ABC~FEDABC~EFDABC~DFEThusABC~FDEABC~EDFABC~ABCInFig.4.1,ifthematchingABC~DEFgivesacongruence,wecanstateiisanidentitycongruence.thatACandDFarecorrespondingsides,andLBCAandLEFDarecorre-iFortheisoscelestriangleRST(Fig.4.4),wherespondingangles.CanyoufindtheotherpairsofcorrespondingsidesandRT==ST,itcanbeshownthat,undertheone-to-onecorrespondingangles?correspondenceRST~SRT,thefigurecanbeRsTwoscalenetrianglescanhaveonlyasingleone-to-onecorrespondencemadetocoincidewithitself.Fig.4.4.whichwillgiveacongruence.Twoisoscelestrianglescanhavetwoone-to-oneExercisescorrespondenceswhichwillgivecongruence.I.Drawa6CH]anda6KLM.ListallthepossiblematchingsofthesecondTMtrianglewiththeorderedsequenceCH]ofthefirsttriangle.2.IfthematchingRST~LMKgivesacongruencebetween6RSTand.6LMK,listallthepairsofcorrespondingsidesandcorrespondinganglesofthetwotriangles.3.Writedownthesixmatchingsofequilateral6ABCwithitself,beginningwiththeidentitycongruenceABC~ABC.4.WritedownthefourmatchingsofrectangleABCDwithitself.5.Inmatching6ABCwith6RST,ACandRTwerematchedascorrespond-ingsides.Doesitthenfollowthat(I)LBandLSarecorrespondingRsKLangles?(2)BCandSTarecorrespondingsides?Fig.4.2.6.Whichofthefollowingfiguresformmatchedpairsthatarecongruenttoeachother?InFig.4.2,ifRT==STandKM==LM,thetwocorrespondencesRST~KLMandRST~LKMmightgivecongruences.WewilldeterminelatewhatadditionalconditionsmustbeknownbeforethetrianglescanbeprovecIcongruenttoeachother.iTheorderinwhichmatchingpairsofverticesaregivenisnotimportan~inexpressingacongruenceandthevertexyoustartwithisnotimportantiInFig.4.3,wecoulddescribetheone-to-onecorrespondenceinonelinea(a)(b)(c)'DEFG~HKjIorEFGD~KjIH.Therearetwoothers.Canyoufin4them?Allthatmattersisthatcorrespondingpointsbematched.Itshouldbeevidentthatatrianglecanbemadetocoincidewithitself.FJ~r:~(d)(e)(l)DEHKFig.4.3.Prob.6.-- 110FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLESIII7-12.IneachofthefollowinguserulerandprotractortofindwhichtrianglesshowninFig.4.5,theshapeofthetriangleseemtobecongruent.Thenindicatethepairsofsidesandanglesinisfixed.Itcannotbechangedwithoutthetriangleswhichseemtomatchinacongruence.bendingorbreakingthepiecesofwood.However,ifwebolttogetherfour(ormore)cDboards,formingafour-sidedfigureasshowninFig.4.6,theshapeoftheframecanbechangedbyexertingaforceononeofthebolts.ThemeasuresoftheanglesformedbytheboardscanbechangedinsizeeventhoughthelengthsofthesidesofABthefigureremainthesame.TheframeofFig.4.5.ABDFig.4.6canbemaderigidbyboltingaEx.7.Ex.8.boardacrossDandF(orEandG),thusformingtworigidtriangles.Therigidityoftrianglesisillustratedinthepracticalapplicationsofthispropertyintheconstructionofmanytypesofstructures,suchasbridges,FEfowers,andgates(Fig.4.7).4.10.Congruenceoftriangles.Theengineerandthedraftsmanarecon-tinuallyusingcongruenceoftrianglesintheirwork.Byapplyingtheirknow-ledgeofcongruenttriangles,theyareabletostudymeasuresofthethreesidesandthethreeanglesofagiventriangleandtocomputeareasoftriangles.ADOftentheyapplythisknowledgeinconstructingtriangularstructureswhichAwillbeexactduplicatesofanoriginalstructure.Ex.9.Ex.lO.Definition:IfthereexistssomecorrespondenceABC~DEFofthever-ticesofL.ABCwiththoseofL.DEFsuchthateachpairofcorrespondingsidesf]arecongruent-amt--eaeh-pair--of-collcspul1diltganglesaleCOllgluellt,tilecorrespondenceABC~DEFiscalledacongruencebetweenthetriangles.Thetrianglesarecongruenttriangles.OrwemaystatethatL.ABCiscon-gruenttoL.DEF,writtenL.ABCDcA==L.DEF.ABDEX.n.Ex.l2.4.17.Thetriangleisarigidfigure.Muchofourstudyofcongruenceofgeometricfiguresdealswithtriangles.Thetriangleisthemostwidelyusedofallthegeometricfiguresformedbystraightlines.Thetriangleisrigidinstructuraldesign.IfthreeboardsareboltedtogetheratA,B,andC,asFig.4.6.-------------- 112FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES1134.19.Basiccongruencepostulate.Althoughwedefinedtwotrianglesascongruentifthreepairsofsidesandthreepairsofanglesarecongruent,trianglescanbeprovedcongruentiffewerpairsofcorrespondingpartsareknowntobecongruent.Wemustfirstacceptanewpostulate.postulate17(theS.A.S.postulate).Twotrianglesarecongruentiftwosidesandtheincludedangleofoneare,respectively,congruenttothetwosidesandtheincludedangleoftheother.Thispostulatestatesthat,inFig.4.9,ifABED,AC====EF,andLA==LE,then!'iABC==!'iDEF.CFA~RDEFig.4.9,Fig.4.7.ThestudentoftenwillfindthatheisaidedinmakingaquickselectionofthecongruentsidesandcongruentanglesinthetwotrianglesbydesignatingThus,if!'iABC==!'iDEF(Fig.4.8),weknowsixrelationshipsbetweenthejthemwithsimilarcheckmarksforthecongruentpairsofcongruentsidesandcongruentangles.InthIStextweWIlttrequentlyllse_ul1ashmarkstoIndI-cate"given"congruences.Thus,inFig.4.10,ifitisgiventhatAC==DE,mAB=mDEAB==DEAB==DB,AC1-ADandDE1-AD,thestudentcanreadilyseewhicharethemBC=mEFBC==EFcongruentpairs.mAC=mDFAC==DFItwillalsobehelpfulif,inprovingamLA=mLDLA==LDcongruencefortwotriangles,thestu-cmLB=mLELB==LEdentnamesthetrianglesinsuchawaymLC=mLFLC==LFastoindicatethematchingvertices.A,'lForexample,inFig.4.10,sinceABC~BTheequationsintheleftcolumnandthecongruencesintherightcolumn!DEEcanbeprovedacongruence,itmeanthesamething.Theycanbeusedinterchangeably.wouldbemoreexplicittorefertotheseEInSection9.2wewillintroduceathirdwaytoindicatecongruencyofsegments."trianglesas"!'iABCand!'iDBE"rather,Flf{.4.1O,cthan,say,"!'iABCand!'iDEB."Al-'thoughthesentence"!'iABC==!'iDEB"canbeprovedcorrect,thesentence"LABC==!'iDBE"willprovemorehelpfulsinceitaidsinpickingoutthe~~correspondingpartsofthetwofigures.ABDE.Itisimportantthatthestudentrecognize,inusingPostulate17toproveFig.4.8.tnanglescongruent,thatthecongruentanglesmustbebetween(formedby) 114FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES115Tthecorrespondingcongruentsides.IfSTATEMENTSREASONSthecongruentanglesarenotbetweentheR~Stwoknowncongruentsides,itdoesnotI.AC==DF;BC==EF.1.Givennecessarilyfollowthatthecorrespon-2.LCandLFareright,1;.2.Given.Ldencewillgiveacongruence.InL.RST3.LC==LF.3.Rightanglesarecongruent.andL.KLM(Fig.4.11)notethat,though4.LABC==L.DEF.4.S.A.S.RS==KM,ST==ML,andLR==LK,thetrianglescertainlyarenotcongruent.C4.20.ApplicationofPostulate17.In4.22.IllustrativeExample1:ThebisectorofthevertexofanisoscelestriangledividesitintotwoPostulate17wehavestatedthattwotri-congruenttriangles.angles,eachmadeupofthreesidesandKthreeangles,arecongruentifonlythreeGiven:IsoscelestriangleABCwithAC==BC;CDbisectsLACB.Fig.4.11.particularpartsofonetrianglecanbeshowncongruentrespectivelytothethreeConclusion:L.ADC==L.BDC.Proo!'correspondingpartsofthesecondtriangle.Hereafter,whenwearegivenanytwotrianglesinwhichweknow,orcanprove,twosidesandtheincludedangleofonetrianglecongruentrespectivelytotwosidesandtheincludedADBangleoftheother,wecanquotePostulate17asthereasonforstatingthattheIllustrativeExample1.twotrianglesarecongruent.STATEMENTSREASONSItisessentialthatthestudentmemorize,orcanstatetheequivalentinhisownwords,thestatementofPostulate17becausehewillberequiredfre-1.AC==BC.I.Given.quentlyinsubsequentproofstogiveitasareasonforstatementsinthese2.CD==CD.2.Reflexivetheoremofsegments.proofs.Afterthestudenthasshowncompetenceinstatingthepostulate,the,j3.CDbisectsLACB.3.Given.instructormaypermithimtoreferbrieflytoitbytheabbreyiationS.A.S.(side-angle-side).Thisabbteviatlc)tiWillbeusecthereafterin[histext:.~4.La==LfJ-.4~A~tor-divides-auan.gkinto.twocongruentangles.OncePostulate17isacceptedastrue,itbecomespossibletoprovevarious5.LADC==L.BDC.5.S.A.S.congruencetheoremsfortriangles.Wewillnextconsideratheoremandtwootherexamplesofhowthispostulatecanbeusedinprovingothercon-4.23.IllustrativeExample2:Cgruences.Given:TheadjacentfigureswithADTheorem4.13andCEbisectingeachotheratB.4.21.IfthetwolegsofonerighttrianglearecongruentrespectivelytotheBConclusion:L.ABC==L.DBE.twolegsofanotherrighttriangle,thetrianglesarecongruent.Given:L.ABCandL.DEFwithACDF,Proo!,IllustrativeExample2.==EBC==EF;LCandLFareright,1;.STATEMENTSREASONSMBC~~;:~m;on6/)m~I.ADandCEbisecteachotheratB.1.Given.2.BA==BD.2.Definitionofbisector.3.BC==BE.3.Reason2.4.LABC==LDBE.4.Verticalanglesarecongruent.5.6ABC~11==L.DBE.5.S.A.S.Theurem4.13.ACD--.-- 116FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES1174.24.Useoffiguresingeometricproofs.Everyvalidgeometricproof~sshouldbeindependentofthefigureusedtoillustratetheproblem.Figuresareusedmerelyasamatterofconvenience.Strictlyspeaking,before1~/CExample2couldbeproved,itshouldbestatedthat:(1)A,B,C,D,andEare'ABp~fivepointslyinginthesameplane;(2)BisbetweenAandD;and(3)BisbetweenCandE.Ex.5.Ex.6.Toincludesuchinformation,whichcanbeinferredfromthefigure,wouldmaketheprooftediousandrepetitious.Inthistextitwillbepermis-sibletousethefiguretoinfer(withoutstatingit)suchthingsasbetweenness,collinearityofpoints,thelocationofapointintheinteriorortheexteriorofanangleorinacertainhalf-plane,andthegeneralrelativepositionofpoints,lines,andplanes.,,07MlZlNThestudentshouldbecarefulnottoinfercongruenceofsegmentsandangles,bisectorsofsegmentsandangles,perpendicularandparallellines.Ex.7.Ex.8.justbecause"theyappearthatway"inthefigure.Suchthingsmustbeincludedinthehypothesesorinthedevelopedproofs.Itwouldnot,forexample,becorrecttoassumeLAandLDarerightanglesinthesecondexamplebecausetheymightlooklikeit.Exercises(A)AxnThetrianglesofeachofthetwelvefollowingproblemsaremarkedtoshowZBcongruentsidesandangles.IndicatethepairsoftriangleswhichcanbeDEprovedcongruentbyPostulate17orTheorem4.13.Ex.9.Ex.10.cF0sIVM~NDAER~rBDsEX.i.Ex.2.RTAccEx.11.EX.i2.FEExercises(B)CEProvethefollowingexercises:A~/~l13.Given:AC.1AB;DE.1BD;A~~nABDAC==DE;BbisectsAD.BEx.3.Ex.4.Conclusion:6.ABC==6.DBE.Ex.13.--------------------------------- 118FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES119nC14.Given:ADandBEintersectingatC;"19.Given:Isosceles,6.ABCwithCE==CB;AC==DC.AC==BC;Dthemid-Conclusion:,6.ABC==,6.DEe.cpointofAC;EtheXmidpointofBe.ABConclusion:,6.ACE==,6.BCD.gABEX.14.Ex.19.Q15.Given:QS..1RT;SbisectsRT.sConclusion:,6.RSQ==,6.TSQ.20.Given:,6.QRSwithLSQR==LSRQ;R~TTthemidpointofQS;SWthemidpointofRS;Ex.15.QS==RS.Conclusion:,6.TQR==,6.WRQ.Dc16.Given:LDAB==LCBA;EA==BF.Conclusion:,6.ABE==,6.BAF.EFQREx.20.Theorem4.14A4.25.Iftwotriangleshavetwoanglesandtheincludedsideofonecon-EX.16.gruenttothecorrespondingtwoanglesandtheincludedsideoftheother,thetrianglesarecongruent.s17.Given:RS==QT,PS==PT;LRTP==LQSP.Conclusion:,6.RTP==,6.QSP.A~BDETTheorem4.14.QEx.17.Given:,6.ABCand,6.DEFwithLA==LD,LB==LE,AB==DE.Conclusion:,6.ABC==,6.DEF.Proofc(3STATEMENTS18.Given:AC==AD;BC==BD;REASONSLa==L(};L{3==Ly.AB1.ABConclusion:,6.ABC==DE,LA==LD.1.Given.==,6.ABD.IJ'Y2.Onl5FthereisapointHsuch$2.Pointplottingpostulate.DthatmDH=mAC.3.DrawHE.Ex.18.3.Twopointsdeterminealine.--n__-------------------------------------- 120FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES1214.,6.ABC:::=,6.DEH.4.S.A.S.i,§ofcongruen~;c5.LDEH:::=LB.5.Corresponding&.are:::=toeachother.I4.27.IllustrativeExample1:6.LB:::=LE.6.Given.Giz'en:CDbisectsLACB;CD..LAB.7.LDEH:::=LE.7.Congruenceof,§istransitive.Conclusion:,6.ADC:::=,6.BDG.8.EliandEFarethesameray.8.Angleconstructionpostulate.Proal'9.H=F.9.Twolinesintersectinatmos"onepoint.10.,6.ABC:::=,6.DEF.10.ReplacingHofStatement4byIllwtrativeExample1.A.[)R(fromStatement9).STATEMENTSREASONSItwillbenotedthatindrawingthefigurefortheproofofTheorem4.14JthepointHisshownbetweenDandF.Thepointcouldjustaswell1.CDbisectsLAGB.1.Given.drawnwithFbetweenHandD.Thiswouldnotalterthevalidityofth2.LACD:::=LBCD.2.Abisectordividesanangleintoproof.TheabbreviationforthestatementofthistheoremisA.S.A.twocongruentangles.3:CD..LAB.3.Given.Theorem4.154.LADCandLBDCarerightangles.4.Twoperpendicularlinesform4.26.Ifalegandtheadjacentacuteangleofonerighttrianglearecon,rightangles.gruentrespectivelytoalegandtheadjacentacuteangleofanother,therig5.LADG:::=LBDG.5.Rightanglesarecongruent.trianglesarecongruent.6.CD:::=CD.6.Congruenceofsegmentsisreflexive.HE7....6ADG:::=,6.BDC.7.A.S.A.Thestudentwillnotehowthemethodofmodusponenshasbeenappliedin~~themLC;"",,/IIFig.4.12.M///'/mLCBE>mLA.///c/Proof/Definition:AsegmentisamedianofaABEtriangleiffitsendpointsareavertexofthe",triangleandthemidpointoftheoppositeTheorem4.17.side.Everytrianglehasthreemedians.ThedottedlinesegmentsofFig.4.13illus-.ABtratemediansofatriangle.Itcanbeshown-MSTATEMENTSREASONSthatthethreemedia?sofatrianglepass,''Fig.4.13.throughacommonpOInt..,'..,,.,1.LetMbethemidpointofBC.1.Everysegmenthasoneandonlyonemidpoint.,12.BM==CM.2.Definitionofmidpoint.Definition:Ananglebisectorofatriangleisasegmentwhichdividesan3.LetDbeapointontheray3.Segmentconstructionpostulate.angleofthetriangleintotwocongruen~nglesandhasitsendpointsona.'vertexandthesideoppositetheangle.BDisthebisectorofLBoff1ABCin.~.oppositeN/A,suchthatMD==i'vIA------Fig.4.14.Everf-ffiaIfgreh~fsthieenangTebIseCtors.IfcanDe-shownthati---4.DrawBD.4.Postulate2.thethreeanglebisectorsmeetinacommonpointwhichisequidistantfromthethreesidesofthetriangle.5.LBMD==LCMA.5.Verticalanglesarecongruent.6.f1BMD==f1CMA.6.S.A.S.7.mLMBD=mLC.7.CorrespondingAiof==&are==.8.rnLCBE=mLMBD+mLDBE.8.Postulate14.cT9.rnLCBE=mLC+mLDBE.9.Substitutionproperty.10.mLCBE>mLC.10.c=a+bAb>0~c>a.AR~;?mLCBEcanbeprovedgreaterthanmLA,inlikemanner,bytakingMasFig.4.14.Fig.4.15.themidpointofABanddrawingeM.Definitions:IfSisbetweenRandQ,thenLQSTisanexteriorangleoff1RST(Fig.4.15).Everytrianglehassixexteriorangles.TheseexteriorTheorem4.18anglesformthreepairsofverticalangles.LRandLTarecallednonadjacent4.35.IftwotriangleshavethethreesidesofonecongruentrespectivelytointerioranglesofLQST.thethreesidesoftheother,thetrianglesarecongruenttoeachother. 132FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES133cFExercises(A).IT1.Given:RT==RU;TS==US.iKProve:(a)~RTS==~RUS.A~//BD~E(b)RSbisectsLTRU.I"""II///Rs//"""I//"I'{-/G'-""uH"...Theorem4.I8.Ex.I.Given:~ABCand~DEFwithAB==DE,BC==EF,AC==DF.Conclusion:~ABC==~DEF.c2.Given:Isosceles~ABCwithProofAC==BC;CDbisectsLACB.STATEMENTSREASONSProve:(a)~ADC==~BDC.(b)AD==BD.1.AB==DE.1.Given.(c)CD1-AB.AB2.ThereisarayAHsuchthat2.Angleconstructionpostulate.DLBAH==LEDF,andsuchthatEx.2.CandGareonoppositesidesofAB.3.ThereisapointGonAHsuch3.Pointplottingpostulate.thatAG3.Given:M==DF.J==KL;4.DrawsegmentBG.4.Postulate2.JL==KM.5.~ABC==~DEF.5.S.A.S.Prove:(a)LM==LL.6.AC==DF.6.Given.(b)LLJK==L?7.AG==AC.7.Theorem3.4.(c)LLKM==L?J~E8.BG==EF.8.CorrespondingEx.3.are==.9.BC==EF.9.Given.10.BG==BC.10.Theorem3.4.11.DrawsegmentCG.II.Postulate2.C12.LACK==LACK.12.Theorem4.16.13.LBCK4.Given:AC==LBCK.13.Theorem4.16.==BC;14.LACB==LACB.14.Angleadditiontheorem.AD==BD.15.LACB==LDFE.15.Reason8.Prove:CDistheperpendicularA~B16.LACB==LDFE.16.Congruenceof~istransitive.bisectorofAB.17.~ABC==~DEF.17.S.A.S.DEx.4. 134FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES135Exercises(B)R~-::.~;:~:;~--~s///////',./9.InthefigureforEx.9itisde-/5.Given:RT==R'T';'/'siredtodeterminethedistance-;;.:::':........PRS/'-.......==R~;RLsp/'.................betweentwostationsRandSon/'medianTP,//'-,==medianT'P'.oppositesidesofabuilding.Ex-Prove:(a)L:.RPTN/'~N==L:.R'p'T'.T'plainhowtwomenwithonlya(b)L:.RST==L:.R'S'T'.Ex.9.tapemeasurecanaccomplishthetask.Proveyourmethod.R'~sPP'_/~-10.Describeameansof,withEx.5.tapeandprotractor,measur--L.=+=ingroughlythedistanceGP.~.--+-Lacrossastream.Provethevalidityofthemethod.6.Given:AE,BD,andFGarestraightlines.~ACEC;.H==DC==Be.A~E11.Provethatthemediantothebaseofanisoscelestriangleequalsthealti-Prove:(a)L:.ABC==L:.EDC.tudetothatbase.(b)L:.AFC==L:.EGC.G12.ProvethatthemedianstothetwocongruentsidesofanisoscelestriangleEx.6.arecongruent.13.Provethat,ifthemedianofatriangleisalsoanaltitudeofthattriangle,thetrianglemustbeisosceles.c14.Provethat,ifapointonthebaseofanisoscelestriangleisequidistant=-fromthenHdpeHHse!tfie'£ffflgt'ttefiHidcs,thepointbisectstilebase.7.Given:IsosceTesL::L4BC~ithAC==BC:15.ProvethattheintersectionoftheperpendicularbisectorsofanytwosidesM,N,P,aremidpointsofAC,BC,ofatriangleareequidistantfromthethreeverticesofthetriangle.andABrespectively.Prove:LAPM==LBPN.ABEx.7.p8.Given:RP==LP;RS==LT;PS==PT.Prove:(a)L:.RTP==L:.LSP.(b)LPSR==LPTL.RLEx.S. 134FUNDAMENTALSOFCOLLEGEGEOMETRYCONGRUENCE-CONGRUENTTRIANGLES135Exercises(B)R~-::.~;:~:;~--~s///////',./9.InthefigureforEx.9itisde-/5.Given:RT==R'T';'/'siredtodeterminethedistance-;;.:::':........PRS/'-.......==R~;RLsp/'.................betweentwostationsRandSon/'medianTP,//'-,==medianT'P'.oppositesidesofabuilding.Ex-Prove:(a)L:.RPTN/'~N==L:.R'p'T'.T'plainhowtwomenwithonlya(b)L:.RST==L:.R'S'T'.Ex.9.tapemeasurecanaccomplishthetask.Proveyourmethod.R'~sPP'_/~-10.Describeameansof,withEx.5.tapeandprotractor,measur--L.=+=ingroughlythedistanceGP.~.--+-Lacrossastream.Provethevalidityofthemethod.6.Given:AE,BD,andFGarestraightlines.~ACEC;.H==DC==Be.A~E11.Provethatthemediantothebaseofanisoscelestriangleequalsthealti-Prove:(a)L:.ABC==L:.EDC.tudetothatbase.(b)L:.AFC==L:.EGC.G12.ProvethatthemedianstothetwocongruentsidesofanisoscelestriangleEx.6.arecongruent.13.Provethat,ifthemedianofatriangleisalsoanaltitudeofthattriangle,thetrianglemustbeisosceles.c14.Provethat,ifapointonthebaseofanisoscelestriangleisequidistant=-fromthenHdpeHHse!tfie'£ffflgt'ttefiHidcs,thepointbisectstilebase.7.Given:IsosceTesL::L4BC~ithAC==BC:15.ProvethattheintersectionoftheperpendicularbisectorsofanytwosidesM,N,P,aremidpointsofAC,BC,ofatriangleareequidistantfromthethreeverticesofthetriangle.andABrespectively.Prove:LAPM==LBPN.ABEx.7.p8.Given:RP==LP;RS==LT;PS==PT.Prove:(a)L:.RTP==L:.LSP.(b)LPSR==LPTL.RLEx.S. I:).Twotrianglesarecongruentiftwosidesandanangleofoneare==respec-tivelytotwosidesandanangleoftheother.4.Twotrianglesthathave==basesand==altitudesarecongruent.5.Thebisectorsoftwoadjacentsupplementaryanglesareperpendiculartoeachother.6.Thebisectorsoftwoanglesofatriangleareperpendiculartoeachother.7.Twoequilateraltrianglesarecongruentifasideofonetriangleis==toasideoftheother.8.Ifthesidesofoneisoscelestriangleare==tothesidesofasecondisoscelestriangle,thetrianglesarecongruent.9.Thealtitudeofatrianglepassesthroughthemidpointofaside.SummaryTests10.Themeasureoftheexteriorangleofatriangleisgreaterthanthemeasureofeitherofthetwononadjacentinteriorangles.11.Anexteriorangleofatriangleisthesupplementofatleastoneinteriorangleofthetriangle.12.Iftwotriangleshavetheircorrespondingsidescongruent,thenthecorre-Test1spondinganglesarecongruent.COMPLETIONSTATEMENTS13.Iftwotriangleshavetheircorrespondinganglescongruent,thenthecorrespondingsidesarecongruent.1.Anangleofatriangleisanangleformedbyonesideofthetr:i14.Notwoanglesofascalenetrianglearecongruent.angleandtheprolongationofanothersidethroughtheircommonpoint.15.Thesidesoftrianglesarelines.2.Aofatriangleisthelinesegmentjoiningavertexandthemid~16.ThereispossibleatriangleRSTinwhichLR=LT.t7:lIlift.')1=6.')1ft,then6H.')1ISeqUIlateral.3.Correspondingsidesofcongruenttrianglesarefoundoppositeth18.Adjacentanglesaresupplementary.anglesofthetriangles.19.Thesupplementoranangleisalwaysanohtllseangle.20.Aperpendiculartoalinebisectstheline.diculartotheoppositeside.21.Themediantothebaseofanisoscelestriangleisperpendiculartothe5.partsofcongruenttrianglesarecongruent.base.6.Thebisectorofthevertexangleofanisoscelestriangleis22.Anequilateraltriangleisequiangular.base.23.Iftwoanglesarecongruenttheirsupplementsarecongruent.7.Theanglesofanisoscelestrianglearecongruent.24.Thebisectorofanangleofatrianglebisectsthesideoppositethatangle.8.Ifthemedianofatriangleisalsoanaltitude,thetriangleis-25.Iftwoisoscelestriangleshavethesamebase,thelinepassingthroughtheir9.Thebisectorsoftwosupplementaryadjacentanglesformaverticesbisectsthebase.10.ThesideofarighttriangleoppositetherightangleiscalledthecTest3Test2EXERCISESTRUE-FALSESTATEMENTSI1.Supplythereasonsforthestatementsin1.Twotrianglesarecongruentiftwoanglesandthesideofonearecan]thefollowingproof:.gruentrespectivelytotwoanglesandthesideoftheother.jGIven:AC==BC'AD==BD.A~B2.IftworighttriangleshavethelegsofonecongruentrespectivelytoProve:AB.1CD:th'twolegsoftheother,thetrianglesarecongruent.11ProofEX.i.D136137----..--------------------------- STATEMENTSREASONS1.AC==BC;AD==BD.1./5/2.LCAE==LCBE;2.LDAE==LDBE.3.LDAC==LDBC.3.4.6DAC==6DBG.4.5.LACE==LBGE.5.6.CE==CEo6.7.6ACE==6BCE.7.8.LAEC==LBEC.8.9....AB..1CD.9.ParallelandperpendicularlinesKT5.1.Parallellines.Parallellinesarecommonplaceintheeverydayexper-iencesofman.Illustrationsofparallellinesaretheyardmarkersonafootballfield,thetopandbottomedgesofthispage,aseriesofverticalfenceposts,andtherailsonwhichthetrainsrun.(SeeFig.5.1.)Parallellinesoccurinanumberofgeometricfigures.TheselineshavecertainpropertiesthatproduceGconsequencesinthesefigures.AknowledgeRoftheseconsequencesisusefultothecraftsman,theartisan,thearchitect,Ex.2.Ex.3.andtheengineer.listaswebeganourstudyofcongruenttriangleswithadefinitionof2.Given:6GJKwith3.Given:Isosceles6RSTwithcongruenttrianglesandwithcertainacceptedpostulates,sowewillbeginHK==IK;GH==IIRT==ST;mediansOIl)studyofparallellineswithadefinitionandapostulate.BymeansofthisProve:GK==JK.SMandRN.definitionandpostulateandthetheoremsalreadyproved,weshallproveProve:SM==RN.severaladditionaltheoremsonparallellines.Definition:Twolinesareparallelifftheylieinoneplaneandwillnotmeet.Thesymbolfor"parallel"or"isparallelto"is"/1".Asamatterofcon-venience,wewillstatethatsegmentsareparallelifthelinesthatcontainthemareparallel.Wewillsimilarlyrefertotheparallelismoftworays,arayandasegment,alineandasegment,andsoon.Thus,inFig.5.2,thestatements;WIiDE,ACl/l2,AC"DE,IIDF,areeachequivalenttothestatementII//l2.Twostraightlinesinthesameplanemusteitherintersect"orbeparallel.However,itispossiblefortwostraightlinesnottointersectandyetnotbeEara]Jeliftheydonotlieinthesameplane.ThefronthorizontaledgeoftheboxofFig.5.3,forexample,willnotintersectthebackverticaledge~(,becausetheydonotlieinthesameplane.Theselinesaretermedskew138139-------~ 140FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES141I~1L~7Fig.5.4.ParallelPlanes.Wehavealreadyprovedthattwolinesareperpendicuiariftheymeettoformcongruentadjacentangles.Perpendicularplanesaredefinedinasimilarway.Definition:Twoplanesareperpendicularifftheyformcongruentad-jacentdihedralangles.PlaneMandplaneN(Fig.5.5)areperpendiculariffLPQS=LPQR.PFig.5.1.ParallelPiPewaysatanoilrefineryRlines.Definitions:Twoplanesareparalleliftheirintersectionisanullset.lineandaplaneareparalleliftheirintersectionisanullset.G..11Fig.5.5.-=-DABCTheorem5.1.....12Theorem5.1DEF5.2.Iftwoparallelplanesarecutbyathirdplane,thelinesofintersectionAareparallel.Fig.5.2.Fig.5.3.Given:PlanePintersectingparallelplanesMandN,withABandCJjtheirIfplanesMandN(Fig.5.4)areparallelwewriteMN.Ifline~2alinesofintersection.IIProve:ABplaneMareparallel,wewritel2UnlesslinesIIandl2ofFIg.IICD.IIMorMIll2'Proof"lieinacommonplane,theyarecalledskewlines.--------------- PARALLELANDPERPENDICULARLINES143142FUNDAMENTALSOFCOLLEGEGEOMETRYSTATEMENTSREASONSTheautomobilemechanicindeterminingwhyanenginewillnotstartmustfirstconsiderthevariouscausesforsuchfailure.Supposeheconcludesthat1.PlanePintersectsplanesMandN1.Given.thefaultmustbeeither(1)nogasolinereachingthecylinderor(2)nosparkinABandCD,respectively.atthesparkplug.Ifhecanshowthatoneofthesedefinitelycannotbethe2.PlaneMIIplaneN.2.Given.fault,hethenconcludesthattheothermustbeandactsonthatbasis.3.ABandCDlieinplaneP.3.Given.ThestUdentmayask,"WhatifIcannotexcludeallbutoneoftheassumed4.ABnCD=O.4.Definitionofparallelplanes.possibilit!es?"Allhecanbecertai~1ofinthatevent~sthathehasnoproof.5.ABIICD.5.Definitionofparallellines.ItispossiblethatoneofthealternativeshehaschosenISactuallytrue.Thereisnoonewaytodeterminewhichalternativestoselectfortestinginanin-5.3.Indirectmethodofproof.Thusfarthemethodsweusedinprovildirectproof.Perhapsseveralexampleswillhelphere.theoremsandoriginalexerciseshavebeendirect.Wehaveconsideredt:Ainformationgivenintheproblem,and,byusingcertainacceptedtruths'5.4.IllustrativeExample1:theformofdefinitions,postulates,andtheorems,havedevelopedalogicstep-by-stepproofoftheconclusion.IthasnotbeennecessarytoassumeGiven:mAB=mBC;mCD0/=mAD.consideroneormoreotherconclusions.Prove:BDdoesnotbisectLABC.However,notalwaysistheinformationcompleteenoughorsufficienlProof"positivetoenableustoreachadefiniteconclusion.Oftenthegivenfatandassumptionsmayleadtotwoormorepossibleconclusions.Itthentccomesnecessarytoknowtheexactnumberofpossibleconclusionswhi,mustbeconsidered.Eachoftheseconclusionsmustbeinvestigatedtermsofpreviouslyknownfacts.Ifallthepossibleconclusionsbut0:IllustrativeExamPle1.DcanbeshowntoleadtoacontradictionorviolationofpreviouslyprovedSTATEMENTSREASONSacceptedfacts,wethencanstatewithauthoritythattheoneremainingmu:beacorrectconclusion.ThismethodiscalledtheindirectmethodofproOf!1.mAB=mBC.I.Given.exclusion.Itisusedextensivelybyallofus.2.mCD7"mAD.2.Given.Supposeyouturnontheswitchtoafloorlampandthelampdoes3.AB==BC.3.Definitionofcongruentseg-light.Howmightyoufindthecauseofthedifficulty?Letusconsiderdments.variouspossiblecausesforfailure.Theymightbe:unscrewedlightbull4.Eitherjj]jbisectsLABCorjjjj4.Lawoftheexcludedmiddle.bulbburnedout,faultywiringinlamp,lampunpluggedinwallsocket,futdoesnotbisectLABC.blownout,nocurrentinyourneighborhood,badwiringinthehous5.AssumeEDbisectsLABC.5.Temporaryassumption.Assumethatincheckingyoufindthatotherlightsinthehousewillburn,t16.LCBD==LABD.6.Definitionofanglebisector.bulbisscrewedinthesocket,thelampisproperlypluggedinthewallsock~.7.BD==BD.7.CongruenceofsegmentsISandthebulbwilllightwhenscrewedinanotherfloorlamp.Bythesete~reflexive.8.6CBDyouhaveeliminatedallbutonepossiblecauseforfailure.Thusyoumuj==LABD.8.S.A.S.concludethatthefailureliesinthewiringinthelamp.9.CD==AD.9.Correspondingpartsofcon-Alawyerfrequentlyusestheindirectmethodofproofinprovinghisdielgruenttrianglesarecongruent.]0.innocentofmisconduct.Letussupposetheclientisaccusedofarm~]mCD=mAD.10.Reason3.1.Statement10contradictsstate-robberyofatheaterat21standMainstreetat7:30p.m.onagivennigtII.Statements10and2.Itisevidentthattheclientwaseither(1)atthatlocalityatthespecifiedtinment2.]2.anddateor(2)hewassomewhereelse.IfthelawyercanprovethatthedieHenceassumption5isfalseand12.Rulefordenyingthealternative.wasatsomeotherspotatthetimeoftherobbery,onlyoneconclusionclBDdoesnotbisectLABC.result.Hisclientcouldnothavebeentherobber.------- 144FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES145C5.5.IllustrativeExample2:3.Acustomerreturnedaclocktothejeweler,claimingthattheclockwouldnotrun.Heofferedasevidencethefactthattheclockstoppedat2:17Given:L.ABCwithCD..lAB;mAC=I'mBC.a.m.afterhisbutlerwounditbeforeretiringafewhoursearlier.WhenProve:mAD=I'mBD.thejewelercheckedtheclockhecouldfindnothingwrongwiththeProoFclockexceptthatitwasrundown.UponwindingtheclockitfunctionedA~'Dproperly.Whatconclusionwouldyoumakeifyouwerethejeweler?IllustrativeExamPle2.4.ThestoryistoldofTomJonesaskingpermissionofthelocaljailkeeperSTATEMENTSREASONStoseeaprisoner.Hewastoldthatonlyrelativeswerepermittedtoseetheinmate.Beingaproudman,Mr.Jonesdidnotwanttoadmithis1.L.ABCwithCD..lAB.1.Given.relationshiptotheprisoner.Hestated,"BrothersandsistershaveI2.mAC=I'mBG.2.Given.none,butthatman'sfatherismyfather'sson."Whereuponthejailer3.EithermAD=mBDormAD=I'3.Lawoftheexcludedmiddle.permittedhimtoseetheprisoner.mBD.Considerthefollowingpossiblerelationshipsbetweentheprisonerand4.AssumemAD=mBD.4.Temporaryassumption.!Mr.Jones:cousin,uncle,father,grandfather,grandson,son,brother.5.AD==BD.5.Definitionofcongruentseg-:IByindirectreasoningdeterminethetruerelationshipbetweenthements.prisonerandMr.Jones.6.CD==CD.6.Congruenceofsegments5.Giveanexampleeitherfromyourownexperienceorahypotheticalcasereflexive.inwhichtheindirectmethodofproofwasused.7.LADCandLBDC'areright7.Perpendicularangles.angles.Exercises(B)8.LADC==LBDC.8.Rightanglesarecongruent.9.L.ADC==L.EDC'.9.S.A.S.Provethefollowingstatementsbyassumingthattheconclusionisnottrue10.AC==BC.10.Correspondingpartsofcon-andthenshowthatthisassumptionleadstoanimpossibleresult.gruenttrianglesarecongruent.6.Ifthemeasuresoftwoanglesofatriangleareunequal,themeasuresof11.mAC=mBC.11.Reason5.,id"0ppo.,ilelhema,'eunequal.ment2.7.Given:mACdoesnotequalmBC;13.Assumption4isfalseandmAD=I'13.Rulefordenyingthealternative.'ICDbisectsLACB.mBD.Conclusion:CDcannotbeperpendicularA~.DtoAB.Exercises(A)j,1Ex.7.1.Tom,Jack,Harry,andJimhavejustreturnedfromafishingtripinJim'sjcar.AfterJimhastakenhisthreefriendstotheirhomes,hediscoversa:;bone-handledhuntingknifewhichoneofhisfriendshasleftinthecar.''''"HerecallsthatTomusedafish-scalingknifetocleanhisfishandthat,,,8.Given:mRTisnotequaltomST;,Harryborrowedhisknifetocleanhisfish.DiscusshowJimcouldreason',o.,,j,MbisectsRS.whoseknifewasleftinhiscar.Indicatewhatassumptionshewouldhave'jtomaketobedefinitelycertainofhisconclusion.Conclusion:TMisnotperpen-2.Twoboyswerearguingwhetherornotasmallanimalintheirpossession'diculartoRS.R~sMwasaratoraguineapig.Whatwasprovediftheboysagreedthatguinea!1pigshavenotailsandtheanimalinquestionhadatail?Ex.8. 146FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES1475.6.Propertiesofexistenceanduniqueness.Thedefinitionofparallel9.Given:mLNoFmMN;linesfurnishesuswithanimpractical,ifnotimpossible,directmethodofPNis..LLM.determiningwhethertwolinesareparallel.WemustresorttotheindirectConclusion:PNdoesnotbisect~iimethod.ButfirstwemustprovetwobasictheoremsaboutperpendicularLLNM.lines.Theseproofsinvolvethepropertiesofexistenceanduniqueness.LpMWehaveassertedtheideaofexistenceanduniquenessinseveralpostulatesandtheoremsinpreviouschapters.ThestudentmayrefreshhismemoryEx.9.onthisbyreferringtopostulates2,3,5,15,16,andTheorems3.1,3.2,3.3.Theexpressions"exactlyone"or"oneandonlyone"meantwothings:C(I)Thereisatleastoneofthethingsbeingdiscussed.(2)Thereisatmostoneofthethingsbeingdiscussed.10.Given:&ABCandA'B'C'withmAB=mA""7JT,mAC=mA"C',A~BStatement(1)aloneleavesopenthepossibilitythattheremaybemorethanmLAoFmLA'.onesuchthing.Statement(2)leavesopenthepossibilitythattherearenoneConclusion:mBCoFmB"C'.ofthethingsbeingdiscussed.Together,statements(1)and(2)assertthereisexactlyonethinghavingthegivenpropertiesbeingdiscussed.A~BTheorem5.21-Ex.10.B+-c5.7.Inagivenplane,throughanypoint//h]ofastraightlinetherecanpassoneand//11.Whatconclusionscanyoudrawifthefollowingstatementsaretrue?onlyonelineperpendiculartotheI1.fJgivenline./~q.P----,2.q~w.IA3.Pistrue.Giue/!.LineI,pointPofI./c=b12.Whatconclusionscandrawnifthefollowingfourstatementsaretrue?//1.fJ~q.Conclusion:m2/2.w~p.1.Thereisalinem]..LIsuchthatPE/tm]3.v~w.m](existence).4.qisfalse.2.Thereisatmostonelinem]..LIsuch13.Giventhefollowingthreetruestatements.thatPEm](uniqueness).Theorem5.2.1.Ifxisa,thenyis{3.2.lExis-y,thenyis8.Proof3.IfYis{3,thenzis1./1.(a)ComPlete:xisa;thenyisandzis-.-(b)Canyoudrawanyconclusionsaboutxifyouknowyis8?STATEMENTSREASONS14.Whatconclusionscanbedrawnfromthefollowingtruestatements?1.Noonecanjointhebridgeclubunlesshecanplaybridge.ProofofExistence:2.Nolobst~rcanplaybridge..LetAbeapointonI..jI.ThereisapointBinhalf-planeh]1.Angleconstructionpostulate.3.NooneISallowedtotalkatthebndgetableunlessheISamemberof;thebridgeclub.JsuchthatmLAPB=90.4.Ialwaystalkatthebridgetable.2.ThenPB(orm])is..LI.2.Definitionof..Llines. 148FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES149ProofofUniqueness:Theorem5.33.Eitherthereismorethanoneline3.Lawofexcludedmiddle.throughPand..1tolorthereis'5.8.Throughapointnotonagivenlinethereisatleastonelineperpen-notmorethanonelinethroughPdiculartothatgivenline.and..1tol.Given:Linel.PointPnotcontainedinl.4.Supposethereisasecondline4.Temporaryassumption.m2..1tolatP.LetCbeapointConclusirJn:AtleastonelinecancontainPandbeperpendiculartol.onm2andinthehalf-planehI,5.mLAPB=90.5.Statement1..~6.mLAPC=90.6.Perpendicularlinesformright~,1/p/themeasureofarightangle=90.]7.Thisisimpossible.7.Statements5and6contradictthe//'~/angleconstructionpostulate.'./._.8.Thiscontradictionmeansthatour,I~/IKR8.Rulefordenyingthealternative./Assumption4isfalse.Hence,//"-"-thereisonlyoneline,satisfying/"-,theconditionsofthetheorem.;/"-T,"-S'",Thecondition"inagivenplane"isanessentialpartofTheorem5.2.Ifwe~didnotstipulate"inagivenplane,"theexistencepartofthetheorem~Theorem5.3.wouldbetrue,buttheuniquenessoftheperpendicularwouldnotbetrue.JFig.5.6illustratesseverallinesperpendiculartoalinelthroughapointofthetProofline.Itcanbeprovedthatallperpendicularstoalinethroughapointon!_~thatlinelieinoneplaneandthatplaneisperpendiculartotheline.The;uniquenessofthisplanecanalsobeproved.1STATEMENTSREASONS1.LetQandRbeanytwopointsofl.1.Postulate2.DrawPQ.2.LPQRisformed.2.Definitionofangle.3.Inthehalf-planeoflnotcontain-3.Angleconstructionpostulate.ingPthereisaray,QJ,suchthatLRQS==LRQP.4.ThereisapointTonQJsuchthat4.Pointplottingpostulate.QT==QP.5.DrawPT.5.Postulate2.6.QK6.Reflexivepropertyofcongru-==QK.ence.Fig.5.6.7.6.PKQ7.S.A.S.==6.TKQ.8.LPKQLTKQ.8.Corresponding,§of==&.are==.Definition:Alinewhichintersectsaplaneinexactlyonepointbutisnot~==9.PT..1l.9.Theorem3.14.perpendiculartotheplaneissaidtobeobliquetotheplane.j 150FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES151Theorem5.4Exercises5.9.ThroughagivenexternalpointIneachofthefollowingindicatewhetherastatementisalwaystrueornotthereisatmostoneperpendiculartoaalwaystrue.givenline.1.Atriangledeterminesaplane.Given:Linel,pointPnotcontainedinl.2.Twoperpendicularlinesdetermineaplane.Conclusion:Nomorethanonelinecanml3.Twoplaneseitherintersectorareparallel.containPandbe1-tol.4.InspacethereisoneandonlyonelinethroughapointPonline1thatQisperpendiculartol.Proof5.Morethanonelineinspacecanbedrawnfromapointnotonline1perpendiculartol.6.Ifline1EplaneM,lineqEplaneM,1nq=P,liner1-line1atP,thenTheorem5.4.liner1-planeM.STATEMENTSREASONS7.Ifline1EplaneM,lineqEplaneM,1nq=P,liner1-l,r1-q,thenr1-M.8.Ifline11-planeM,thenplaneM1-linel.1.Eitherthereismorethanone1.Lawofexcludedmiddle.9.IfPEplaneM,thenonlyonelinecontainingPcanbeperpendiculartolinethroughP1-to1orthereisM.notmorethanonelinethrough10.IfPEplaneM,LAPBandLCPBarerightangles,andmLAPC=91,P1-tol.thenPB1-M.2.Assumemland~aretwosuch2.Temporaryassumption.linesandintersecting1atAandBBrespectively.3.Ontherayopposite;rpcon-3.Pointplottingpostulate.structAQsuchthatAQ~AP.4.DrawQB.4.Postulate2.5.AB==AB.5.Congruenceofsegmentspcreflexive.6.LPABandLQABarerightAi.6.Theorem3.13.A7.LPAB==LQAB.7.Theorem3.7.8.LPAB==LQAB.8.S.A.S..9.LQBA==LPBA.9.CorrespondingAiof&,are~.iEx.lO.==11.Anynumberoflinescanbedrawnperpendiculartoagivenlinefroma10.LPBAisarightangle.10.Perpendicularlinesformrighlangles.pointnotontheline.12.Whenweprovetheexistenceofsomething,weprovethatthereisexactly11.LQBAisarightangle.11.Substitutionproperty.12.BQ1-l.12.Definitionofperpendiculoneobjectofacertainkind.lines.13-18.Inthefigure(Exs.13-18),planeMd-planeN,planeMnplaneN=13.Statements12and2contradict13.StatementsRS,AGliesinM,BDliesinN,ACnlID=P.Theorem5.2.5.2.13.RS1-AC:14.Assumption2mustbefalse;then14.RulefordenyingthealternatiVI14.liJj1-AG.thereisatmostoneperpen-15.mLAPB=90.dicularfromPtol.16.mLCPD=90. 152FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES15321.IfPA==QAandPC==QC,thenACistheperpendicularbisectorofPQ.22.IfPA==QAandPA==BA,thenDA==BA.23.IfL-.DB-Q==L-.DBP,thenPA==QA.B24.IfDCistheperpe'1dicularbisectorofPQ,thenPD==QD..§.IITheorem5.5IAI5.10.Iftwolinesinaplaneareiperpendiculartothesameline,theyIIareparalleltoeachother.mIPRGiven:mandnarecoPlanar,m..1I,n..1I;n=:::::.Conclusion:mIIn.ProofExs.13-18.Theorem5.5.17.IIlLCPDisthemeasureofadihedralangle.18.LBPSandLBPRareadjacentdihedralangles.STATEMENTSREASONS19-24.Inthefigure(Exs.19-24),A,B,C,DarepointsinplaneM;Pc!nMc.1.mandnarecoplanar,m..1I,n..1I.1.Given.19.PCandClIdetermineauniqueplane.2.EitherIIIIInorm%n.2.Lawofexcludedmiddle.20.PJjandDQdetermineauniqueplane.3.AssumeIII%n.3.Temporaryassumption.4.mandnmustmeet,sayatP.4.Nonparallelcoplanarlinesinter-psect.5.Thenmandnaretwolinespass-5.Statements1and3.ingthroughanexternalpointand..1tothesameline.6.Statement5contradictsTheorem6.Statements5andTheorem5.4.5.4.11/IInistheonlypossibleconclu-Eitherpornot-p;not(not-p)~fl.sionremaining.Theorem5.65.11.Twoplanesperpendiculartothesamelineareparallel.GlVen:PlaneM..1lineI;planeN..1lineI.COnclusion:PlaneMIIplaneN.Q(ThistheoremisprovedbytheindirectmethodofproofandisleftasaneXerciseforthestudent.)Ex.I.19-24. 154FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES1555.13.Theparallelpostulate.Havingprovedtheexistenceofalinethroughanexternalpointandparalleltoasecondline,itwouldseemthatthenextstepwouldlogicallybetoproveitsuniqueness.Strangeasitmayseemat...........first,thiscannotbedoneifwearetouseonlythepostulateswehavestated............,A......thusfar.Wemustassumethisuniquenessasapostulate.......I.......I............postulate18(theparallelpostulateorPlayfair'spostulate).............*Througha....-:::givenpointnotonagivenlinethereisatmostonelinewhichcanbedrawnparallel""'.-.-tothegivenline.'>-""""............ThusinFig.5.7,ifinaplaneweknowthat.....RS~pS'isparalleltoABandpassesthroughP,weR;IB--I'-mustassumethat,ifCDpassesthroughP,I'CDcannotbeparalleltoAB;or,ontheother,,;..ABhand,ifCDisparallelton,CDcannotpass~Fig.5.?throughP.Postulate18wasassumedbyEuclid.SincethattimemanymathematicianshavetriedtoproveordisprovethispostulatebymeansofotherpostulatesTheorem5.6.andaxioms.Eacheffortmetwithfailure.Asaconsequence,mathe-maticianshaveconsideredwhatkindofgeometrywouldresultifthispropertyTheorem5.7werenotassumedtrue,andseveralgeometriesdifferentfromtheonewhich5.12.Inaplanecontainingalineandapointnotontheline,thereisat.wearestudyinghavebeendeveloped.Suchageometryisknownasnon-leastonelineparalleltothegivenline.Euclideangeometry.Given:LinelwithpointPnotcontainedtDuringthenineteenthcenraryNicholasLobachevsky(1793-1856),aRussianinl.mlmathematician,developedanewgeometrybaseduponthepostulatethatConclusion:IntheplaneofPandlthereIthroughagivenpointtherecanheanynumberoflinesparalleltoagivenline.~~~~~In1854astilldifferentnon-EuclideangeometrywasdevelopedbyBern-isatleastonelineItthatcanbedrawnthroughPandIhardRieman(1826-1866),aGermanmathematician,whobasedhisdevelop-Iparalleltol.mentontheassumptionthatalllinesmustintersect.AgeometrysomewhatPdifferentfromanyofthesewasusedbyAlbertEinstein(1879-1955)inIProofIdevelopinghisTheoryofRelativity.IThesegeometriesarequitecomplex.EuclideangeometryismuchsimplerIIandservesadequatelyforsolvingthecommonproblemsofthesurveyor,thetContractor,andthestructuralengineer.Theorem5.7.Theorem5.8STATEMENTSREASONS~-----5.14.Twolinesparalleltothesame:::>ePm~1.Pisapointnotonlinel.1.Given.lineareparalleltoeachother.2.LetmbealinethroughPand..12.Theorem5.3.Given:IIIn,mIIn.tol.Conclusion:111m.n3.LetIIbealinethroughPinthe3.Theorem5.2.ProofplaneoflandPand..1tom.Theorem5.8.4.IIIII.4.Theorem5.5.*ThisstatementisattributedtoJohnPlayfair(1748-1819),brilliantScottishphysicistandmathe-matician.--------------- 156FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES157STATEMENTSREASONSTheorem5.105.16.Alineperpendiculartooneoftwoparallelplanesisperpendicularto1.IIIrz,mIIrz.1.Given.theother.2.EitherIIImorI,.Jrm.2.Lawoftheexcludedmiddle.3.Assume1,./1'm.3.Temporaryassumption.4.Thenlandmmeetat,sayP.4.Twononparallellineslyinginthesameplaneintersect.5.Thenlandmpassthroughthe5.Statements1and3.samepointandareparalleltothesameline.Given:PlaneMisparalleltoplaneN;JiBisperpen-6.Thisisimpossible.6.Postulate18.diculartoplaneM.7.:.lllm.7.Rulefordenyingthealternative..;Conclusion:JiBisperpendicu-EitherPOI'not-p;not(not-p)~p.lartoplaneN.Theorem5.9Proof----1--~:,.-~5.15.InaplanecontainingtwoparallelPImlines,ifalineisperpendiculartooneofIC----DIthetwoparallellinesitisperpendicular~----NIItotheotheralso.IIEInIIIGiven:mIIn,lintheplaneofmandn,l.1n.A.-----Conclusion:I.1m.ProofJTheorem5.9.Theorem5.10.STATEMENTSREASONSSTATEMENTSREASONS1.rn/In;lliesinplaneofmandn.1.Given.I.PlaneM1.Given.2.I.1n(orn.1I).2.Given.(DefinitionsareIIplaneN;JiB.1planeM.ible.)3.Eitherl.1mol'lisnot.1tom.3.Lawofexcludedmiddle.2.ThroughJiBpassplaneRinter-2.Postulate5.4.Assumelisnot.1tom(ormisnot4.Temporaryassumption.sectingplanesMandNinGFandDC,respectively;alsothroughAiJ.1tol).5.Thenthereisalinemjinthe5.Theorem5.2.passplaneSintersectingplanesMplaneofmandnthatis.1tolatandNinGF!andDE,respectively.thepointPwheremintersectsi.3.CFIIDeandGF!15E.3.Theorem5.1.6.Thenmj6.Theorem5.5.4.AB.1GF:,AB.1~-.:r4.DefinitionofperpendiculartoIIrl.7.Thisisimpossible.7.Postulate18.plane.8.:.l.1m.8.Rulefordenyingthealternative.5.AB.1DC;Ai!.1DE.5.Theorem5.9.6.A1f.1planeN.6.Reason4.EitherPOI'not-p;not(not-p)~p.------------------------------------------------------- 158FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES1595.17.Transversalsandspecialangles.STATEMEKTSREASONSAtransversalisalinewhichintersectstwoormorestraightlines.InFig.5.8,1.La==Lf3.1.Given.tisatransversaloflineslandm.When2.EitherIIImorI%m.2.Lawoftheexcludedmiddle.twostraightlinesarecutbyatrans-;).AssumeI%m.3.Temporaryassumption.versal,eightanglesareformed.4.Thenlmustmeetm,say,atP.4.NonparallellinesinaplanemustTherearefourangleseachofwhichisintersect.asubsetoftUl.Twoofthese(Lx5.6RSPisformed.5.Definitionofatriangle.andLw)containbothAandB.There6.Lf3isanexteriorLof6RSP.6.DefinitionofexteriorLofa6.arealsofourangleseachofwhichis7.Lf3>La.7.Theorem4.17.asubsetoftUm.Twoofthese(LsFig.5.8.8,Thisisimpossible.8.Statements1and7conflict.andLk)containbothAandB.Thei9.:./llm.9.Rulefordenyingthealternative.fourangles(Lx,Lw,Ls,Lk)whichcontainbothAandBarecalledinteriorjnot(not-p)~p.angles.Theotherfour(Ly,Lz,Lr,Lq)arecalledexteriorangles.IThepairsofinterioranglesthathavedifferentverticesandcontainpoints!onoppositesidesofthetransversal(suchasLsandLworLxandLk)are:1calledalternateinteriorangles.iTheorem5.12ThepairsofexterioranglesthathavedifferentverticesandcontainpointsI5.19.Iftwostraightlinesarecutbyaonoppositesidesofthetransversal(suchasLrandLzorLqandLy)ardtransversalsoastoformapairofcon-calledalternateexteriorangles.Jgruentcorrespondingangles,thelinesareCorrespondinganglesareapairconsistingofaninteriorangleandaniparallel.exterioranglewhichhavedifferentverticesandlieinthesameclosedhalf-jplanedeterminedbythetransversal.Examplesofcorrespondinganglesare;.Given:LineslandmcutbymLqandLw.TherearefourpairsofcorrespondinganglesinFig.5.8.transversalt;La==Ly.Sincewewillusetheterm"transversal"onlywhenthelineslieinonepiaConclusion:111m.wewillnotrepeatthisfactineachofthefollowingtheorems.ProofTheorem5.11Theorem5.12.5.18Iftwostraightlinesformcongruentalternateinteriorangleswhenthey!arecutbyatransversal,theyareparallel.STATEMENTSREASONS1.La==Ly.1.Given.2.Ly==Lf3.2.Verticalanglesare==.3.LaGiven:Lineslandmcutby==Lf3.3.CongruenceofanglesistransI-transversaltatRandS;tive.aV4.:.La==Lf3.IIIm.4.Theorem5.11.Conclusion:IIIm.m!":JQProof5.20.Corollary:IftwolinesarecutbyatransversalsoastoforminteriorS.upplementaryanglesinthesameclosedhalf-planeofthetransversal,theTheorem5.11.hnesareparallel.(Theproofofthiscorollaryislefttothestudent.) 160FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES161Exercises5.Given:LM==NT,LT==NM..1.InthefollowingfigurelistthepairsProve:TNofanglesofeachofthefollowingIILM.I~Ntypes.(a)Alternateinteriorangles.Ex.5.(b)Alternateexteriorangles.(c)Correspondingangles.(d)Verticalangles.CF(e)Adjacentangles.6.Given:A,D,B,Earecollinear;BC==EF;AD==BE;AC==DF.Ex.I.Prove:BCIIEF.A~'Ex.6.2.Inthefigure,iftheanglesareofthemeasuresindicated,whichlineswouldnkbeparallel?7.Given:m;n1-l;k1-m.m3l"Prove:n"k.m92°Ex.2.Ex.7.C3.Given:BisthemidpointofAEandDCECD.A~BProve:ACIIDE.~Ex.3.[]ABEx.8.Ex.9.4.Given:RTandPSarediagonals;PQ==QS;N18.AcollapsibleironingboardisconstructedtothatthesupportsbisectRQQT.==eachother.Showwhytheboardwillalwaysbeparalleltothefloor.Prove:PTIIRS;9.Thedraftsmanfrequentlyusesadevice,calledaparallelruler,todrawRPIIST.RSparallellines.TherulerissoconstructedthatAB==DCandAD==BC.Ex.4.Thepinsattheverticespermittherulertobeopeneduporcollapsed. 162FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES163IflineABissuperimposedonagivenlinem,theedgeDCwillbeparallelTheorem5.13tom.Showwhythisistrue.10.Adraftsmanfrequentlydrawstwo5.21.IftwoparallellinesarecutbyalinesparallelbyplacingastraightBDtransversal,thealternateinterioranglesmedge(T-square)rigidatadesiredarecongruent.pointonthepaper.HethenslidesacelluloidtrianglewithbaseflushGiven:mIIn;transversaltcuttingnatBnwiththestraightedge.Withtri-andmatA.anglesinpositionsIandIIheisConclusion:La==Lf3.thenabletodrawlineABIIlineCD.Why?Ex.10.ProofTheorem5.13.II.Given:K,L,Marecollinear;QSTATEMENTSREASONSKL==NL,mLMLN=mLK+mLN;I.miln.I.Given.LQbisectsLMLN.rQa{32.EitherLa==Lf3orLaisnot==2.LawoftheexcludedmiddleProve:IIKN.KULf3.LM3.AssumeLaisnot==Lf3.3.Temporaryassumption.EX.n.4.LetIbealinethroughAforwhich4.Angleconstructionpostulate.thealternateangles'arecon-gruent,i.e.,La==Ly.5.ThenIIIn.5.Theorem5.11.6.Thisisimpossihle.6.StatementsIand5contradictPostulate18.12.Given:AC==BC'DC==Ee.Prove:DEIIAB.'(Hint:DrawCG7....La==Lf3.7.Rulefordenyingthealternative.inamannerwhichwillhelpyourproof.)Theorem5.145.22.Iftwoparallellinesarecutbyatrans-Aversal,thecorrespondinganglesarecongruent.EX.i2.(Theproofofthistheoremisleftasanexerciseforthestudent.)cDDFC13.Given:EFbisectsDCandAB;LA==RLB;AD==Be.Theorem5.14.Prove:DCTheorem5.15IIAB.(Hint:UseTheorem5.5)A/bB5.23.Iftwoparallellinesarecutbyatransversal,theinterioranglesontheEsamesideofthetransversalaresupplementary.(TheproofisleftasanEX.i3.exerciseforthestudent.) PARALLELANDPERPENDICULARLINES165164FUNDAMENTALSOFCOLLEGEGEOMETRY6.Given:BAIIDC;A(BExercisesmLB=40;1.Given:ABIICiJ;mLBPD=70./fEIIEF.Find:thenumberofdegreesinProve:La==Ly.ABLD.(Hint:Drawauxiliary:J~+linethroughP<1CTS==PS.RProve:AO==CO;Prove:RS==QS.DO==BO.Ex.9.Ex.4.C"AD:10.Provethatalinedrawnparalleltothebase5.Given:RSIIPQ;ofanisoscelestriangleandthroughits0ismidpointofAB.~~vertexwillbisecttheexteriorangleattheProve:0ismidpointofCD.Acvertex.Ex.5.Ex.lO. 166FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES16711.Given:LM==TN;LMTheorem5.16CIITN..D-flProve:LTIIMN.~N5.24.Themeasureofanexteriorangleofatriangleisequaltothesumofthe~/(rneasuresofthetwononadjacentinter-ABELAtiorangles.Theorem5.16.Given:LCBEisanexteriorLof6ABe.EX.n.Conclusion:mLCBE=mLC+mLA.Proof1DC/STATEMENTSREASONS/12.Given:/11m;AD1-m;BC1-m.///Prove:AD/==Be.//I.DrawBJjIIAe.I.Postulate18;Theorem5.7.m/AB2.mLcx=mLC.2.Theorem5.13.3.mLf3=mLA.3.Theorem5.14.Ex.12.4.mLCBE=mLcx+mLf3.4.Angleadditionpostulate..5.mLCBE=mLC+mLA.5.Substitutionpropertyofequality.13.Given:AD==BC;AD1-AB;BC1-AB.5.25.Sumoftheanglemeasuresofatriangle.AcceptanceoftheparallelProve:DC==ABandDCIIAB.postulatemakespossibletheproofofthenexttheorem,oneofthemost:~:widelyusedtheoremsindealingwithfiguresinaplane.Youareprobablyfamiliarwithit,havinglearnedaboutitinductivelybymeasuringinothermathematicscourses.'Venowproceedtoproveitdeductively.Ex.13.Theorem5.17D~C14.Given:ABIIDC;5.26.ThesumofthemeasuresoftheADIIBC;anglesofatriangleis180.~~LBADisarightLGiven:6ABCisanytriangle.Prove:AC==BD.A~BConclusion:mLA+mLACB+ABmLB=180.Theorem5.17.Ex.14.ProofSTATEMENTSREASONS15.Given:LM==MN==TN==LT.NNl.6ABCisanytriang-l~.Prove:LN1-TM.1.Given.2.ThroughCdrawDEIIAB.2.Theorem5.7;Postulate18.3.mLcx=mLA,mLf3=mLB.3.Theorem5.13.LAt4.mLDCE=mLcx+mLACE.4.Postulate14.5.rnLACEEx.15.=mLACB+mLf3.5.Postulate14. PARALLELANDPERPENDICULARLINES169168FUNDAMENTALSOFCOLLEGEGEOMETRY3.Given:mLA=70;c6.mLDCE=mLa+mLACB6.Substitutionpropertyofequality.mLC=80.+mLf3.Find:mLx=-.7.mLDCE=mLA+mLACB7.Substitutionpropertyofequality.+mLB.~8.mLDCE=180.8.Definitionofstraightangle.AB9.mLA+mLACB+mLB=180.9.Theorem3.5."DEx.3.Itshouldbeevidentthatthesumofthemeasuresoftheanglesofatriangle']dependsonourassumingtrueEuclid'spostulatethatonlyonelinecanbe]drawnthroughapointparalleltoagivenline.Asamatterofinterest,noo-1Euclideangeometryprovesthesumofthemeasuresofthethreeanglesof..I4.Given:mLC=110;triangledifferentfrom180.InthiscoursewewillagreewithEuclid,sinceii~A~mLCBD=155.willprovesatisfactoryforallourneeds.BD"Find:mLa=-.Theproofstothefollowingarelefttothestudent.Ex.4.5.27.Corollary:Onlyoneangleofatrianglecanbearightangleor~Jobtuseangle."'i5.28.Corollary:Iftwoanglesofonetrianglearecongruentrespectivelyto~twoanglesofanothertriangle,thethirdanglesarecongruent.5.Given:AC==BC;5.29.Corollary:Theacuteanglesofarighttrianglearecomplementary.AC1-Be.Find:mLA=-.ALBExercisesEx.5.DeterminethenumberofdegreesintherequiredanglesinExs.1througItJ8.'BB6.Given:mLA=50;1.Given:ABmLB=60;IIED;EmLBAE=50;La==Lf3.AmLDCE=40.Find:mLa=DFind:mLa+mLf3=-.CDEx.6.Ex.I.Cc7.Given:La==La';2.Given:ABLf3==Lf3';==BC==Ae.mLD=130.Find:mLA=-.Find:mLC=-.ABABEx.7.Ex.2.--------------------------- 170FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES171CC8.Given:AC~AB:AC1-All13.Given:STIIAC;Find:mLx=-.SRIIBC;La~L{3.A~.,.A~BProve:LA~LB.SBEx.8.Ex.13.DEc9.Given:AE1-AC;CD1-AC;La~L{3.14.Given:CE1-AB.Prove:LE~LD.AcDB1-Ae.BProve:LDCE~LEBD.Ex.9.ABcEx.14.10.Given:LABCwithLCDE~LB.Prove:LCED~LA.ABAB15.Given:ABCD;IIEx.10.FGbisectsLRFE;EGbisectsLlJJ.c'F.cDEProve:EG1-GF.11.Given:LA~LB;Ex.15.DE1-AB.Prove:La~LE.ADcEx.ll.16.Given:AC~BC;CDF1-AC;EF1-Be.12.Given:BC1-AC;Prove:LAFD~LBFE.DC1-AB.Prove:La~L{3.A~RDAFBEx.12.Ex.16. 172FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES1734.Lx~Ly.4.§5.28.Corollary..5.CD~CD.5.Reflexivepropertyofcongruence.6.liADC~liBDC.6.A.S.A.17.Given:CE1-E;7.AC~BC.7.Correspondingpartsofcon-DE1-AD.gruenttrianglesarecongruent.Prove:LA~LE.A5.31.Corollary:Anequiangulartriangleisequilateral.Theorem5.195.32.IftworighttriangleshaveahypotenuseandanacuteangleofoneEx.17.Econgruentrespectivelytothehypotenuseandanacuteangleoftheother,thetrianglesarecongruent.EGiven:liABCandliDEFwithLBandLEright~;AC~DF;LA~LD.18.Given:AC~BC;FC~EC.Conclusion:liABC~liDEF.Prove:DE1-AB.cFEx.18.ADABTheorem5.18Theorem5.19.Proo}:5.30.Iftwoanglesofatrianglearecongruent,thesidesoppositethemarecongruent.STATEMENTSREASONSGiven:liABCwithLA~LB.1.LBandLEareright~.1.Given.Conclusion:AC~BC.2.LB~LE.2.Rightanglesarecongruent.A3.LA~LD.3.Given.Proof4.LC~LF.4.§5.28.Corollary.Theorem5.18.5.AC~DF.5.Given.STATEMENTSREASONS6.liABC~liDEF.6.A.S.A.I.liABCwithLA~LB.1.Given.2.DrawCDbisectingLC.2.Ananglehasoneandonly00';1Theorem5.20raywhichbisectsit.:,15.33.Iftworighttriangleshavethehypotenuseandalegofonecongruent3.Lm~Ln.3.Abisectordividesanangleinti":itothehypotenuseandalegoftheother,thetrianglesarecongruent.twocongruentangles.------------------------- 174FUNDAMENTALSOFCOLLEGEGEOMETRYjPARALLELANDPERPENDICULARLINES175---1Given:Right6ABC;right6DEF;LCandLFareright,1,;BC~EF,AB=="Theorem5.21cDE.5.34.IfthemeasureofoneacuteangleConclusion:6ABC~6DEF.1.,ofarighttriangleequals30,thelengthofthesideoppositethisangleisone-BEhalfthelengthofthehypotenuse.TheIproofofthistheoremisleftasanIIexerciseforthestudent.(Hint:Ex-IIItendABtoD,makingmBD=mAB.IIDrawCD.ProveCBbisectsADofIIequilateraI6ADG.)IADIAaLDcFTheorem5.21.Theorem5.20.cExercisesProof1.Given:CD1-AB;STATEMENTSREASONSBC1-AC;Prove:LA~LBCD.B1.OntherayoppositeFJjconstruct1.Pointplottingpostulate.FesuchthatFG~CA.A2.DrawEG.2.Postulate2.3.LDFEisarightL.3.Given.Ex.1.4.EF1-GD.4.Definitionofperpendicular.ivJ----..-.----.S-..LGFEKaTighLL.:'~.!.L.:D..rn_njrion.llf:..LLI:n'Trsil00__n-------------------------6.LCisarightL.6.Given.7.LC~LGFE.7.Rightanglesarecongruent.2.G;ven:TM1-LM;8.BC~EF.8.Given.TK1-LK;9.6ABC~6GE1".9.S.A.S.TM~TK.TL10.AB~GE.10.CorrespondingsidesofProve:TLbisectsLKLM.are~.11.AB~DE.11.Given.K12.GE~DE.12.Theorem3.5.13.LGFDisastraightL.13.Definitionofstraightangle.Ex.2.14.LG~LD.14.Baseanglesofanisoscelesb-are~.15.6GE1"~6DEF.15.Theorem5.19.3.Given:AD1-DC;16.FG~FD.16.Reason10.17.CA~FD.17.Theorem3.5(fromBC1-DC;1and16).MisthemidpointofDC;:V~:18.6ABC~6DEF.18.S.A.S.orS.S.S.AM~BM.Prove:AD~Be.Ex.3.--------------------------------------------------------------------- 176FUNDAMENTALSOFCOLLEGEGEOMETRYPARALLELANDPERPENDICULARLINES1774.Given:SQbisectsRLatT;8.Given:CD..DIIBE;RS1-SQ;LQ1-SQ.R~AC1-CD;Prove:RLbisectsSQatT.QmLA=60.Es~Prove:m(AB)=tm(AE).L9.Prove:IfthebisectorofanexteriorangleofatriangleisparalleltoEx.4.theoppositeside,thetriangleisisosceles.CE10.Prove:ThebisectorsofthebaseanglesofanisoscelestriangleintersectatapointequidistantfromtheABC5.Given:AE1-BC;endsofthebase.Ex.8.CD1-AB;II.Prove:AnypointonthebisectorofanangleisequidistantfromthesidesAE==CD.A!;;>Boftheangle.Prove:BA==BC.Ex.5.s6.Given:L,M,R,Tarecollinear;LTRS1-LS;LM==TR;NM1-TN;LL==LT.Prove:RS==MN.NEx.6.TRs7.Given:RT==ST;RS1-TQ.Prove:RQSQ.==QEx.7.------------------------------- 16.Twoplanesareperpendicularifftheyformcongruentadjacentangles.17.Twoplanesperpendiculartothesamelineare18.Throughapointoutsideaplane(howmany?)linescanbedrawnparalleltotheplane.19.1'0righttrianglecanhavea(n)angle.20.Twolinesperpendiculartothesameplanearetoeachother.21.Thetwoexterioranglesatavertexofatriangleareanglesandarethereforeangles.22.ThegeometrywhichdoesnotassumePlayfair'spostulateissometimescalledgeometry.SummaryTestsTest2TRUE-FALSESTATEMENTSTest11.Anisoscelestrianglehasthreeacuteangles.COMPLETIONSTATEMENTS2.Alinewhichbisectstheexteriorangleatthevertexofanisoscelestriangle1.Thesumofthemeasuresoftheanglesofanytriangleisisparalleltothebase.2.Anglesinthesamehalf-planeofthetransversalandbetween3.Themedianofatriangleisperpendiculartothebase.linesare4.Iftwolinesarecutbyatransversal,thealternateexterioranglesare3.Twolinesparalleltothesamelinearetoeachother.supplementary.4.Themeasureofanexteriorangleofatriangleisthanthemeasure5.Theperpendicularbisectorsoftwosidesofatriangleareparalleltoeachofeithernonadjacentinteriorangle.other.5.Aproofinwhichallotherpossibilitiesareprovedwrongiscalled6.Inanacutetrianglethesumofthemeasuresofanytwoanglesmustbe0'-b'~'6.Alinecuttingtwoormorelinesiscalleda(n)7.Ifanytwoanglesofatrianglearecongruent,thethirdangleiscongruent.7.Iftwoisoscelestriangleshaveacommonbase,thelinejoiningtheir8.Iftwoparallelplanesarecutbyathirdplane,thelinesofintersectionareverticesistothebase.skewlines.8.Alineparalleltothebaseofanisoscelestrianglecuttingtheothersides9.Toprovetheexistenceofsomething,itisnecessaryonlytoprovethatcutsoffantriangle.thereisatleastoneofthethings.9.Atriangleisiftwoofitsaltitudesarecongruent.10.Theacuteanglesofarighttrianglearesupplementary.10.Thestatementthatthroughapointnotonagivenlinethereisoneandj11.Theexpressions"exactlyone"and"atmostone"meanthesamething.onlyonelineperpendiculartothatgivenlineassertstheand12.Twoplanesperpendiculartothesameplaneareparallel.]propertiesofthatline.3.Aplanewhichcutsoneoftwoparallelplanescutstheotheralso.]11.Theacuteanglesofarighttriangleare4.Twolinesperpendiculartothesamelineareparalleltoeachother.]5.Twolinesparalleltothesamelineareparalleltoeachother.12.Iftwoparallellinesarecutbyatransversal,theinterioranglesonthesame.sideofthetransversalare.]6.Twolinesparalleltothesameplaneareparalleltoeachother.]13.Ifthesumofthemeasuresofanytwoanglesofatriangleequalsthe:7.Twolinesskewtothesamelineareskewtoeachother.18.Anexteriorangleofatrianglehasameasuregreaterthanthatofanymeasureofthethirdangle,thetriangleisa(n)triangle.:14.Iffromanypointofthebisectorofananglealineisdrawnparallelto:interiorangleofthetriangle.]9.Iftwolinesarecutbyatransversal,thereareexactlyfourpairsofalter-onesideoftheangle,thetriangleformedisa(n)triangle.J15.Twoplanesareiftheirintersectionisanullset.!nateinterioranglesformed.178j179 SUMMARYTESTS181180FUNDAMENTALSOFCOLLEGEGEOMETRY[[.1n.[,m,andnarethreelinessuchthat.1mandm.1n,then20.If21.Anexteriorangleofatriangleisthesupplementofatleastoneinteriorangleofthetriangle.22.Inarighttrianglewithanacuteanglewhosemeasureis30,themeasure,ofthehypotenuseisone-halfthemeasureofthesideoppositethe30.angle.50°~23.Whentwoparallellinesarecutbyatransversalthetwointeriorangleso~thesamesideofthetransversal[arecomplementary.Prob.7.Prob.8..1n,thenn.1m.24.If[,m,andnarelines,[11m,[,p..25.If[,m,n,andparelines,[IIm,nP.1m,andn#-p,thennII.1[passesthroughPandisparalleltolinemifandonlyifPELan<26.LineTest4[nm=.0.[atAandlinematB,thentnEXERCISES27.Iftransversaltintersectsline{A,B},whereA#-B.1.Supplythereasonsforthestatementsinthefollowingproof:Test3DPROBLEMS125°1-8.SolveformLo::Given:AC==BC;CD==CEoProve:DF.1AB.Prob.I.111m;rlls.ABFlEx.I.40°STATEMENTS58°REASONSm~0'Prob.4.1.ACProb.3.111m.==BC;CD==CEo1.2.mLA=rnLB:mLCDE=2.mLCED.3.mLAFD=mLFEB+mLB.3.4.mLFEB=mLCED.4.5.mLFEB=mLCDE.5.6.:.mLAFD=mLCDE+mLA.6.7.8.mLAFD+mLCDE+mLA=180.7.9.mLAFD+rnLAFD=180.8.mLAFDLh10.=90.9.:.l5F----LAB.10.Prob.6.Prob.5.-------------------------- L/61RTBEx.3.Ex.2.Polygons-Parallelograms3.Given:DBbisectsLADC;2Given:RS==LS;SPbisectsLTSL.BDIIAE.:Prove:SPIIRL.Prove:~ADEisisosceles::6.1.Polygons.Manyman-madeandnaturalobjectsareintheshapeofpolygons.Weseepolygonsinourbuildings,thewindows,thetileonourfloorsandwalls,theflag,andtheordinarypencil.Manysnowflakesunderamicroscopewouldberecognizedaspolygons.Thecrosssectionofthebeehoneycombisapolygon.Figure6.1illustratesvariouspolygons.Definitions:Apolygonisasetofpointswhichistheunionofsegmentssuchthat:(I)eachendpointistheendpointofjusttwosegments;(2)notwosegmentsintersectexceptatanendpoint;and(3)notwosegmentswiththesameendpointarecollinear.Thesegmentsarecalledsidesofthepolygon.Theendpointsarecalledverticesofthepolygon.Adjacentsidesofthepoly-gonarethosepairsofsidesthatshareavertex.Twoverticesarecalledadjacentverticesiftheyareendpointsofthesameside.Twoanglesofapolygonareadjacentanglesiftheirverticesareadjacent.Alessrigorousdefinitionforapolygoncouldbethatitisaclosedfigurewhosesidesaresegments.Ifeachofthesidesofapolygonisextendedandtheextensionsintersectnootherside,thepolygonisaconvexpolygon.Figure6.1a,b,c,d,eillustrateconvexpolygons.Figure6.ifillustratesapolygonthatisnotconvex.Inthistextwewillconfineourstudytoconvexpolygons.6.2.Kindsofpolygons.Apolygoncanbenamedaccordingtothenumberofitssides.Themostfundamentalsubsetofthesetofpolygonsisthesetofpolygonshavingtheleastnumberofsides-thesetoftriangles.Everypolygonofmorethanthreesidescanbesubdivided,byproperlydrawingsegments,intoasetofdistincttriangles.182183 184FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMSL~D185(a)(b)(c)Fig.6.3Exterioranglesofapolygon.(d)(e)(f)sides;A,E,C,D,Eareverticesofthepolygon;LA,LB,LC,LD,LEaretheFig.6.1.Polygons.anglesofthepolygon.TherearetwodiagonalsdrawnfromeachvertexofDefinitions:Apolygonisaquadrilateraliffithasfoursides;itisapentthefigure.goniffithasfivesides,ahexagoniffithassixsides,anoctagoniffithaseighlDefinition:Anexteriorangleofapolygonisananglethatisadjacentto.sides,adecagoniffithastensides,andann-goniffithasnsides.andsupplementarytoanangleofthepolygon(Fig.6.3).Definitions:Apolygonisequilateraliffallitssidesarecongruent.-c6.3.Quadrilaterals.Unlikethetriangle,thequadrilateralisnotarigidpolygonisequiangulariffallitsanglesarecongruent.Apolygonisaregultfigure.Thequadrilateralmayassumemanydifferentshapes.Somepolygoniffitisbothequilateralandequiangular.quadrilateralswithspecialpropertiesarereferredtobyparticularnames.Definitions:ThesumofthemeasuresofthesidesofapolygoniscalledtliWewilldefineafewofthem.perimeterofthepolygon.Theperimeterwillalwaysbeapositivenumbe'Definitions:Aquadrilateralisatrapezoid(symbolD)iffithasoneandAdiagonalofapolygonisasegmentwhoseendpointsarenonadjaceonlyonepairofparallelsides(Fig.6.4).Theparallelsidesaretheb~~esvenice:,ufLhepUlygoll.ThesideUPUllwhichLhepulygonappcaL5-tQ(Upperandlower)ofthetrapezoid.Thenonparallelsidesarcthelegs.TiIeiscalledthebaseofthepolygon.InFig.6.2,ABCDEisapolygonofaltitudeofatrapezoidisasegment,asDE,whichisperpendiculartooneofDthebasesandwhoseendpointsareelementsofthelinesofwhichthebasesaresubsets.Oftenthewordaltitudeisusedtomeanthedistancebetweenthe1/bases.Themedianisthelinesegmentconnectingthemidpointsofthenon-/parallelsides.Anisoscelestrapezoidisonethelegsofwhicharecongruent/cI(Fig.6.5).Apairofangleswhichshareabaseiscalledbaseangles.I--/,/--,/I/--/~dO(~----/'/'DUpperbaseCI'"v~~-//Q__r-T)//Ef::--,/I//Median"""'-J...I------"""'//I/"1,/L~./Base""'"LowerbaseR"""'AsAB'BFig.6.2.Diagonalsandbaseofapolygon.Fig.6.4.Fig.6.5.---------------------------------------------------------------- 186FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS187Dc1.Thesidesofpolygonsaresegments.Definitions:Aquadrilateralisa2.Theoppositesidesofatrapezoidareparallel.parallelogram(symbolD)iffthepairs3.Everyquadrilateralhastwodiagonals.ofoppositesidesareparallel.Anyside4.Sometrapezoidsareequiangular.oftheparallelogrammaybecalledthe5.Allrectanglesareequiangular.base,asABofFig.6.6.ThedistanceBase6.Thesetofparallelogramsaresubsetsofrectangles.betweentwoparallellinesistheperpen-AE7.Anoctagonhaseightangles.diculardistancefromanypointononeFig.6.6.8.Anoctagonhasfivediagonals.ofthelinestotheotherline.An9.Thesetofdiagonalsofagiventriangleisanullset.altitudeofaparallelogramisthesegmentperpendiculartoasideofthe10.Thediagonalsofapolygonneednotbecoplanar.parallelogramandwhoseendpointsareinthatsideandtheoppositesideor11.Everypolygonhasatleastthreeangles.thelineofwhichtheoppositesideisasubset.DEofFig.6.6isanaltitudeof12.Ifapolygondoesnothavefivesidesitisnotapentagon.0ABCD.Here,again,"altitude"isoftenreferredtoasthedistancebe-13.Arhombusisaregularpolygon.tweenthetwoparallelsides.Aparallelogramhastwoaltitudes.]4.EachexteriorangleofapolygonissupplementarytoitsadjacentangleofArhombusisanequilateralparallelogram(Fig.6.7).thepolygon.Arectangle(symbolD)isaparallelogramthathasarightangle(Fig.6.7).15.Onlyfiveexterioranglescanbeformedfromagivenpentagon.Arectangleisasquareiffithasfourcongruentsides.Thusitisanequi-]6.Asquareisarectangle.lateralrectangle(Fig.6.7).QT]7.Asquareisarhombus.DCG]8.Asquareisaparallelogram.]Hp9.Arectangleisasquare.20.Arectangleisarhombus.2].Arectangleisaparallelogram.E22.Aquadrilateralisapolygon.0RectangleABRs23.Aquadrilateralisatrapezoid.Square24.Aquadrilateralisarectangle.RhombusFig.6.7.25.Apolygonisaquadrilateral.Theorem6.16.4.Allanglesofarectanglearerightangles.Exercises(B)Given:ABCDisarectanglewithLAarightangle.Prove:LB,LC,andLDarerightangles.].Drawaconvexquadrilateralandadiagonalfromonevertex.Determine(Theproofofthistheoremislefttothestudent.Hint:UseTheorem5.15.)thesumofthemeasuresofthefouranglesofthequadrilateral.2.DrawaconvexpentagonandasmanydiagonalsaspossiblefromoneofD--Citsvertices.(a)Howmanytrianglesareformed?(b)WhatwillbetheSumofthePleasuresoftheanglesofthepentagon?3.Repeatproblem2forahexagon.4.Repeatproblem2foranoctagon.5.Usingproblems2-5asaguide,whatwouldbethesumofthemeasures.oftheanglesofapolygonof102sides?AB6.Whatisthemeasureofeachangleofaregularpentagon?Theorem6.1.Exercises(A)7.Whatisthemeasureofeachangleofaregularhexagon?8.Whatisthemeasureofeachexteriorangleofaregularoctagon?Indicatewhichofthefollowingstatementsarealwaystrueandwhichare.9.Whatisthemeasureofeachangleofaregulardecagon?notalwaystrue.n-------------------------------------- 188FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS18910.UsingthesetofpolygonsastheUniversalset,drawaVenndiagram'Theorem6.3DCrelatingpolygons,rhombuses,quadrilaterals,andparallelograms.6.11.Thediagonalsofaparallelogram11.UsingthesetofquadrilateralsastheUniversalset,drawaVenndiagrambisecteachother.~relatingquadrilaterals,squares,parallelograms,rhombuses,andtrape-Given:DABCDwithdiagonalsintersectingzoids.atE.12.UsingthesetofparallelogramsastheUniversalset,drawaVenndiagramA~BConclusion:ACandBDbisecteachother.relatingparallelograms,squares,rectangles,andrhombuses.ProofTheorem6.3.Theorem6.2DsSTATEMENTSREASONS6.5.Theoppositesidesandtheopposite"",,"'"y,anglesofaparallelogramarecongruent./-//I.ABCDisaD.1.Given.-_//-"",-""2.AB/-IIDe.2.DefinitionofaD.Given:DABCD.-X--3.Lz==Ly,Lr==Ls.3.Theorem5.13.Conclusion:ABAr4.AB==DC;AD==BC;==DC.4.Theorem6.2.LA==LC;LB==LD.Theorem6.2.5.LABE==LCDE.5.A.S.A.Proof6.AE==EC,andBE==DE.6.Correspondingsidesof&,are==STATEMENTSREASONS7.ACandBDbisecteachother.7.Definitionofbisector.6.12.Corollary:Thediagonalsofarhombusareperpendiculartoeach1.ABCDisaD.other.2.DrawthediagonalAC.3.ABDC;ADIIIIBe.Exercises(A)4.Lx==Ly;Lr==Ls.5.ACCopythechartbelow.Thenputcheckmarks(x)wheneverthepolygon==AC.haslileil}dicat~(lrela~911~Jl_U:>nnn_n--6.LABC==LCDA.7.AB==DC;AD==BC.AllsidesOpposssiteDiagonalsbisectOppositeDiagonalsarearesidesareeachthe.1of.1are8.LB==LD.Rclationships--IIotherpolygon.1--9.LAParallelogram==LC.Rectangle6.6.Corollary:Eitherdiagonaldividesaparallelogramintotwocongruenttriangles.Rhombus6.7.Corollary:Anytwoadjacentanglesofaparallelogramaresupple-jmentary.Square6.8.Corollary:SegmentsofapairofparallellinescutoffbyasecondpairTrapezoidofparallellinesarecongruent.6.9.Corollary:Twoparallellinesareeverywhereequidistant.Isoscelestrapezoid6.10.Corollary:Thediagonalsofarectanglearecongruent. 190FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS191QTxercises(B)DC-...-6.Given:RS==QT;...----1.Given:ABCDisa0;.-------RSIIQT....---DR1..AC;BT1..AC..--Prove:QRSTisaD..--IProve:DR==BT.A~RSEX.i.Ex.6.TDFC7.Given:ABCDisaD;~DEbisectsLD;2.Given:QRSTisa0;BFbisectsLB.RM==NT.Prove:DEProve:QMIIBF.~==SN.R~ABEEx.2.Ex.7.CTs3.Given:ABCDisa0;DE1..AB;CF1..ABproduced.ZL-8.ProvethatthediagonalQSofProve:DE==CF.ALlEBFrhombusQRSTbisectsLQandLS.QnEx.3.Ex.8.9.Provethatifthebaseanglesofatrapezoidarecongruent,thetrapezoidCisisosceles.D]0.Provethatifthediagonalsofaparallelogramareperpendiculartoeach4.Given:ABCDisanisoscelesother,theparallelogramisarhombus.]].trapezoidwithAD==Be.Provethatifthediagonalsofaparallelogramarecongruent,itisaProve:LA==LB.rectangle.(Hint:DrawCEIIDA.)AiEDB]2.Provethatthebisectorsoftwoconsecutiveanglesofaparallelogramareperpendiculartoeachother.Ex.4.Theorem6.4DCDC6.13.Iftheoppositesidesofaquad-y---...-,/'~ilateralarecongruent,thequadrilaterals,-----ISaparallelogram.~~~~~~~~~~..-------5.Given:AB==CD;.-...-...-..-r....-----AD./x==BC.-..-Given:QuadrilateralABCDwithABAProve:ABCDisaD.A----==BCD;AD==BC.Theorem6.4.Prove:ABCDisaD.Ex.5.-- 192FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS193ProofSTATEMENTSREASONSSTATEMENTSREASONS1.ACandBDbisecteachotheratE.1.Given.2.AE==CE;BE==DE.2.Definitionofbisector.1.AB==CD;AD==BC.1.Given.3.Lx==Ly.3.Verticalanglesarecongruent.2.DrawdiagonalAe.2.Postulate2.4.LABE==LCDE.4.S.A.S.3.AC==AC.3.Reflexivepropertyofcongruence.5.AB==CD.5.Correspondingpartsof==&.are4.LABC==LCDA.4.S.S.S.congruent.5.Lx==Ly;Lr==Ls.5.Correspondingpartsof==&.are6.Lr==Ls.6.Sameas4.7.ABCD.7.Theorem5.11.6.AB6.Theorem5.11.8.ABCD"isaD.IICD;ADIIBC.8.Theorem6.5.7.:.ABCDisaD.7.DefinitionofD.Theorem6.7m6.16.IfthreeormoreparallellinescutTheorem6.5offcongruentsegmentsononetrans-6.14.Iftwosidesofaquadrilateralare!""-/.versal,theycutoffcongruentsegmentsncongruentandparallel,thequadrilateral////Joneverytransversal.//-isaparallelogram.//Given:Parallellinest,m,andnxIGiven:QuadrilateralABCDwithAcutbytransversalsrandAB==CD;ABIICD.s;ABTheorem6.5.==BC.Conclusion:ABCDisaD.Conclusion:DEI==EF.Prn°f"T.STATEMENTSREASONSSTATEMENTSREASONSTheproofislefttothestudent.1.ThroughDandEdrawDGIIrI.Postulate18;Theorem5.7.andEH2.DCIIr.IIEH.2.Theorem5.8.3.ADIllfECF.3.Given.Theorem6.64.:ADGBandBEHC"arem.4.DefinitionofD.5.AB6.15.Ifthediagonalsofaquadrilateralbisect==DGandBC==EH.5.Theorem6.2.DC6.ABeach0ther,thequadrilateralisaparallelogram.s!==Be.6.Given.7.DGy==EH.7.Theorem3.5andtransitivepro-Epertyof'congruence.Given:QuadrilateralABCDwithAC8.La==L{3andL"y==LB.8.Theorem5.14.andBD-bisectingeachotheratE.x9.LDCE[;jjQ==LEHF.9.~5.28.Conclusion:ABCDisaD.ArB10.LDGE>'==LEHF.10.A.S.A.II.DE==EF.11.§4.28.ProofTheorem6.6.-------------------.-------- 194FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS19.f1ExercisesPN6.Given:LMNPisaD.1.Given:ABCDisaD;PR1-LN;MismidpointofAD;MS1-LN.[S<;JNismidpointofBe.f::::ICProve:RMSPisaD.LMProve:MBNDisaD.ABEx.6.Ex.I.ST7.Given:,6.ABCwithDmidpointCofAC;EmidpointofBC;DE==EF.D~EF2.Given:QRIIST;Prove:ABFDisaD.Lx==Ly.L8.InEx.7,provernDE=tmAB.Prove:QRSTisaD.QAI~IRBEx.2.Ex..7,8.Dc9.Provethattwoparallelogramsarecongruentiftwosidesandtheincludedangleofonearecongruentrespectivelytotwosidesandtheincluded3.Given:ABCDisaD;angleoftheother.AM==CN.Prove:MBNDisaD.A&JlEx.3.10.Given:0QRSTwithAQ==SC;~'4.Given:QRSTisaD;RB==DT.QLbisectsLTQR;T~Prove:ABCDisaD.RSBSMbisectsLRST.Prove:QLSMisaD.SREx.10.Ex.4.11.Given:TrapezoidABCD5.Given:0ABCDwithdiagonals~CwithABintersectingatE.IIDC;AD==De.Prove:EbisectsFG.ABProve:ACbisectsLA.GAMBEx.5.Ex.ll. 196FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS197c12.ProvethatlinesegmentsdrawnH/fromAandBofLABCtotheop-//positesidescannotbisecteachother.//LGK(Hint:UseindirectmethodbyBA/L-_-assumingASandRBbisecteach//other;thenABSRisaD,ete.)A~B//Ex.12.ELD13.Provethataquadrilateralisarhombusifthediagonalsbisecteachotherandareperpendiculartoeachother.Theorem6.8.14.ProvethatiffromthepointwherethebisectorofanangleofatrianglemeetstheoppositesideparallelstotheothersidesaredrawnarhombusGiven:LABCandLDEFwithisformed.'l3tIIEPandwithsamedirection;6.17.Directionofrays.TworayshavethesamedirectionifandonlyifeitherBAEfjandwithsamedirection.IItheyareparallelandarecontainedinthesameclosedhalf-planedetermined:Conclusion:LABC==LDEF.bythelinethroughtheirendpointsorifonerayisasubsetoftheother(Fig.Proof'6.8).STATEMENTSREASONSI1.ExtendB1and8.Labeltheir1.ArayhasinfinitelengthInone;,.~intersectionG.direction.i'~2.BtII8.2.Given..;;;;;;;;;;;;;;;3.L/lBeLK(;H.3.Theorem5..l4.....L-,..==.4.BAilED.4.Given..5.LDEFLK(;H.==5.Theorem5.14.6.LABC==LDEF.6.Theorem3.4.;,.~\Theorem6.9Fig.6.9.Fig.6.8.6.19.IftwoangleshavetheirsidessoTworayshaveoppositedirectionsifandonlyifeithertheyareparallelandmatchedthattwocorrespondingsides,Yarecontainedinoppositeclosedhalf-planesdeterminedbythelinethroughhavethesamedirectionandtheother--LKtheirendpointsorarecollinearandtheintersectionoftheraysisapoint,twocorrespondingsidesareoppositelyAB§../---/----/segment,oranullset(Fig.6.9).directed,theanglesaresupplementary.Given:LABCandLDEFwithlTheorem6.8BCIIEFandwiththesamedirection;ED6.18.Iftwoangleshavetheirsidessomatchedthatcorrespondingsides;BAIIEDandwithoppositedirections.havethesamedirections,theanglesarecongruent.Conclusion:LABCandLDEFaresupplementary.Theorem6.9. 198FUNDAMENTALSOFCOLLEGEGEOMETRYPOLYGONS-PARALLELOGRAMS19918.mAB=mMN+mMN.18.Theorem3.5andsubstitutionProofproperty.19.mA1N+mMN=mAB.19.Symmetricpropertyofequality.STATEMENTSREASONS20.mMN=tmAB.20.Divisionpropertyofequality.Theproofisleftasanexerciseforthestudent.Theorem6.11Theorem6.10cc6.21.Alinethatbisectsonesideofatri-6.20.Thesegmentjoiningthemidpointsofangleandisparalleltoasecondsidetwosidesofatriangleisparalleltothebisectsthethirdside.L/)~thirdside,anditsmeasureisonehalftheAmeasureofthethirdside.Given:MNbisectsAC,RBMN/iAB.Given:6ABCwiththeMthemidpointofConclusion:M1VbisectsBe.Theorem6.11.ACandNthemidpointofBC.AProofConclusion:MNIIAB,mMN=tmAB.Theorem6.10.ProofSTATEMENTSREASONSSTATEMENTSREASONSTheproofisleftasanexerciseforthestudent.1.OntherayoppositeNMcon-1.Pointplottingpostulate.IstructNDsuchthatND~MN.2.DrawBD.2.Postulate2.ITheorem6.12c3.NisthemidpointofBC.3.Given.6.22.Themidpointofthehypotenuseofaright4.NB~Ne.4.Definitionofmidpoint.triangleisequidistantfromitsvertices.5.LDNB~LMNC.5.Verticalanglesare~.6.6DNB~6MNC.6.S.A.S.Given:Right6ABCwithLABCarightL;7.BD~CM.7.Correspondingsidesof~&areMisthemidpointofAe.-.Conclusion:AM~BM~CM.8.MisthemidpointofAC.8.Given.Proof9.CM~AM.9.Reason4.NA10.BD~AM.10.Theorem4.3.11.LDBN~LMCN.11.Reason7.STATEMENTSREASONSTheorem6.12.12.BD12.Theorem5.11.IIAC.13.ABDMisaD.13.Theorem6.5.14.MN14.DefinitionofaD.,Theproofisleftasanexerciseforthestudent.(Hint:DrawMNIIAB.IIAB.Thenprove6BMN~6CMN.)15.MD~AB.15.TheoppositesidesofaCJare16.mMD=mAB.16.DefinitionExercisesments.17.mMD=mMN+mND.17.Postulate1.Given:6ABCwithR,S,TmidpointsofAC,BC,andABrespectively;perty. POLYGONS-PARALLELOGRAMS201200FUNDAMENTALSOFCOLLEGEGEOMETRYcNmAC=6inches;mBC=8inches;mAB=12inches.MFind:thevalueofmRS+/o~:6.Given:QuadrilateralKLMNwithP,Q,mST+mRT.R,SthemidpointsofKL,ATB.1LM,MN,andNK.EX.i.Prove:PRandQSbisecteachother.LBEx.6.c7.Given:ABCDisaD;K,L,M,NaremidpointsofOC,OD,OA,2.Given:LCisarightL;mLA=60;OB.MisthemidpointofAB,Prove:KLMNisaD.mAC=8inches.Find:thevalueofmAB.AEx.7.A8.Given:MNisthemedianoftrapezoidEx.2.ABCD;ACandBDaredia-3.Given:ABIIDC;Misthemid-Dgonals.pointofAD;Nisthemid-Prove:MNbisectsthediagonals.pointofBC.8MBProve:1'1NII4B;lUNIioc;mMN=t(mAB+mCD).AEx.8.(Hint:Drawi5NuntilitmeetsABat,say,P.)4.InthefigureforEx.3,findtheExs.3,4.clengthofABifmDC=8feet,9.Given:BNandAMaremediansofmMN=11feet.LABC;ListhemidpointofOA;KisthemidpointofOB.Prove:KMNLisaD.5.Given:QuadrilateralABCDwithQ,R,10.Provethatthelinejoiningthemid-DS,P,themidpointsofAB,pointsoftwooppositesidesofaBC,CD,andDArespectively.parallelogrambisectsthediagonalofProve:PQRSisaD.theparallelogram.(Hint:Youwillneedtodraw11.Provethatthelinesjoiningthemid-thediagonalsofABCD.)PointsofthesidesofarectangleformABarhombus.Ex.9.-------------------- Test2TRliE-FAI.SESTATEMENTS1.Thediagonalsofaparallelogrambisecteach2.Aquadrilateralthathastwoandonlytwoparallelother:sIdesISarhombus..3.Thebasesofatrapezoidareparalleltoeachother.4.Anequilateralparallelogramisasquare.5.Thediagonalsofaparallelogramarecongruent.6.Anequiangularrhombusisasquare.SummaryTests7.Ifapolygonisaparallelogram,ithasfoursides.8.Thediagonalsofaquadrilateralbisecteachother.9.AparaJlelogramisarectangle.10.Thediagonalsofarhombusareperpendiculartoeachother.11.ThemeasureofthelinesegmentjoiningthemidpointsoftwosidesofaTest1triangleisequaltothemeasureofthethirdside.12.IftwoangleshavetheirCopythefollowingchartandthenplaceacheckmarkinthespaceprovide,correspondingsidesother,theyareeitherrespectivelyifthefigurehasthegivenproperty.congruentorparalleltoeach13.IfthediagonalsofaparallelogramsUpplementary.rectangle.arecongruent,theparallelogramisa14.IfthediagonalsofaparallelogramIisasquare.areperpendicular,ParallelogramItheparallelogramRectangleISquare/Rhombus!15.Aparallelogramisdefinedasaquadrilateralarccongruent.theoppositesidesofwhichBothpairsofoppositesidesare16.Ifthediagonalsofaquadrilateralparallel.areperpendicularquadrilateralisaparallelogram.toeachother,the17.ThenonparaJlelsidesofanisos,."L.Rothpairsof°PIY""..rnase.aRecongruentcongruent.IS.Thelineangles'egmen"joiningIherilateralbisecteachother.midpoint'ofBothpairsofoppositeanglesare°PIMite,id"ofaqnad.19.Iftwosidesofaquadrilateralarecongruent,itisaparallelogram.congruent.20.The.medianofatrapezoidbisectseachdiagonal.21.fhediagonabofaDiagonalsareofequallength.22.Theline,pa'aHeJogramdivideitintofourcong.u",tItiangle,.Ihmughthevettice,ofapacaHeJogTamDiagonalsbisecteachother.formanotherpacaHeJ10Ihediagonal,23.Iftheparallelogram.diagonal,ofa"eetangleaceDiagonalsareperpendicular.ISasquare.Petpendicular,2'.'rh,'hepacaHeiogtamAllsidesarecongruent.'egment'joiniogtheformacon'0Nishalfthedifferenceofthemeasuresroftheinterceptedarcs.DEx.5.Ex.6.7.36.Corollary:ThemeasureoftheangleformedbytwotangentsdrawnAfromanexternalpointtoacircleis80°halfthedifferenceofthemeasuresoftheinterceptedarcs.ABCorollary7.36.AExercisesFindthenumberofdegreesmeasureinLa,Lf3,andinarcs.0isthecenterof,Bacircle.'cEx.7.Ex.8.CDiIIAB.pEx.I.Ex.2.DDscpRDEx.3.Ex.4.Ex.9.Ex.10.~ 240FUNDAMENTALSOFCOLLEGEGEOMETRYCIRCLES241DApc0lC),,j0Lf')pApEx.17.Ex.IS.Ex.11.Ex.12..i,',',.aI.-...'"',--AEx.20.Ex.l3.Ex.14.IpcEx.19.AiF'cEx.15.Ex.16.l Test2TRUE-FALSESTATEMENTSI.Ifaparallelogramisinscribedinacircle,itmustbearectangle.2.Doublingtheminorarcofacirclewilldoublethechordofthearc.3.Onasphere,exactlytwocirclescanbedrawnthroughtwopointswhicharenotendsofadiameter.4.Anequilateralpolygoninscribedinacirclemustbeequiangular.5.Aradiusofacircleisachordofthecircle.6.Ifaninscribedangleandacentralanglesubtendthesamearc,theSummaryTestsmeasureoftheinscribedangleistwicethemeasureofthecentralangle.7.Astraightlinecanintersectacircleinthreepoints.8.Arectanglecircumscribedaboutacirclemustbeasquare.9.Theangleformedbytwochordsintersectinginacircleequalindegreestohalfthedifferenceofthemeasuresoftheinterceptedarcs.10.Atrapezoidinscribedinacirclemustbeisosceles.Test1]I.Allthepointsofaninscribedpolygonareonthecircle.12.Anglesinscribedinthesamearcaresupplementary.COMPLETIONSTATEMENTS13.Alineperpendiculartoaradiusistangenttothecircle.14.Theangleformedbyatangentandachordofacircleisequalindegrees1.In00thediameterAOBandtangentATare-.toone-halfthemeasureoftheinterceptedarc.2.Acentralangleofacircleisformedbytwo-'15.Thelinejoiningthemidpointofanarcandthemidpointofitschord3.Aninscribedangleofacircleisformedbytwo-'isperpendiculartothechord.4.Anangleinscribedinasemicircleisa(n)~angle.16.fheanglebisectorsofatrianglemeetinapointthatisequictisranrfrofh5.Thegreatestnumberofobtuseanglesaninscribedtrianglecanthethreesidesofthetriangle.IS_.17.Twoarcsarecongruentiftheyhaveequallengths.6.Tangentsegmentsdrawntoacirclefromanoutsidepointare_.18.Iftwocongruentchordsintersectwithinacircle,themeasurementsofthe7.Thelargestchordofacircleisthe-ofthecircle.segmentsofonechordrespectivelyequalthemeasurementsofthe8.Anangleisinscribedinanarc.Iftheinterceptedarcisincreasedbysegmentsoftheother.theinscribedangleisincreasedby19.Thelinesegmentjoiningtwopointsonacircleisasecant.-'9.Theoppositeanglesofaninscribedquadrilateralare-.20.Anangleinscribedinanarclessthanasemicirclemustbeacute.10.Alinethroughthecenterofacircleandperpendiculartoachord-21.Theangleformedbyasecantandatangeiltintersectingoutsideacircleisthechordanditsarc.measuredbyhalfthesumofthemeasuresoftheinterceptedarcs.11.Ifalineis-toaradiusatitspointonthecircle,itistangentto22.Iftwochordsofacircleareperpendiculartoathirdchordatitsendpoints,circle.theyarecongruent.oft12.Iftwocirclesintersect,thelinejoiningtheircentersisthe-~3.Anacuteanglewillinterceptanarcwhosemeasureislessthan90.commonchord.Ach~rdofa~ircleisa.diameter.2:.13.Inacircle,orincongruentcircles,chordsequidistantfromthecenterThemtersectwnofalmeandaCirclemaybeanempty.set.26.circleare_.Spheresarecongruentifftheyhavecongruentdiameters.27.14.Anangleformedbytwotangentsdrawnfromanexternalpoint1.If.aplaneandaspherehavemorethanonepointincommon,thesecircleisequalindegreestoone-halfthe-ofitsinterceptedarcS.POintslieonacircle.242243 Test3181PROBLEMSFindthenumberofdegreesinLa,Lf3,andsineachofthefollowing:Proportion-SimilarPolygonsProb.2.Prob.1.8.1.Ratio.Thecommunicationofideastodayisoftenbaseduponcom-paringnumbersandquantities.Whenyoudescribeapersonasbeing6feettall,youarecomparinghisheighttothatofasmallerunit,calledthefoot.Whenapersondescribesacommodityasbeingexpensive,heisreferringtothecostofthiscommodityascomparedtothatofothersimilarordifferentcommodities.Ifyousaythatthedimensionsofyourlivingroomare18by24feet,apersoncanjudgethegeneralshapeoftheroombycomparingthedimensions.WhenthetaxpayeristoldthathiscitygovernmentisspendingProb.4.Prob.3.100°c!~percentofeachtaxdollarforeducationpurposes,heknowsthat42centsOlltorevery100centsareusedforthispurpose.Thechemistandthephysicistcontinuallycomparemeasuredquantitiesinthelaboratory.Thehousewifeiscomparingwhenmeasuringquantitiesofingredientsforbaking.Thearchitectwithhisscaledrawingsandthemachinedraftsmanwithhisworkingdrawingsarecomparinglengthsoflines105°inthedrawingswiththeactualcorrespondinglengthsinthefinishedproduct.Definition:Theratioofonequantitytoanotherlikequantityisthequo-Prob.6.Prob.5.tientofthefirstdividedbythesecond.Itisimportantforthestudenttounderstandthataratioisaquotientofmeasuresoflikequantities.Theratioofthemeasureofalinesegmenttothat35°ofananglehasnomeaning;theyarenotquantitiesofthesamekind.Wecanfindtheratioofthemeasureofonelinesegmenttothemeasureofasecondhnesegmentortheratioofthemeasureofoneangletothemeasureofasecondangle.However,nomatterwhatunitoflengthisusedformeasuringtwosegments,theratiooftheirmeasuresisthesamenumberaslongasthesameunitisusedforeach.Inlikemanner,theratioofthemeasuresoftwo245244 46FUNDAMENTALSOFCOLLEGEGEOMETRYPROPORTION-SIMILARPOLYGONS247(b)3inchesto2feet?nglesdoesnotdependupontheunitofmeasure,solongasthesameunitis(c)3hoursto15minutes?sedforbothangles.Themeasurementsmustbeexpressedinthesame.(d)4degreesto20minutes?nits.:).Maryis5yearsand4monthsold.Hermotheris28yearsand9monthsAratioisafractionandalltherulesgoverningafractionapplytoratios.old.WhatistheratioofMary'sagetohermother's?Iewritearatioeitherwithafractionbar,asolidus,divisionsign,orwiththe4.Whatistheratioofthelengthsoftwolineswhichare7feet8inchesandymbol:(whichisread"isto").Thustheratioof3to4isto3/4,3--;-4,or3:4.,4feet4incheslong?the3and4arecalledtermsoftheratio.:J.Whattwocomplementaryangleshavetheratio4:I?Theratioof2yardsto5feetis6/5.Theratioofthreerightanglestotwo'6.Whattwosupplementaryangleshavetheratio1:3?straightanglesisfoundbyexpressingbothanglesintermsofacommonunit7.GearAhas36teeth.GearBhas12teeth.Whatistheratioofthe(suchasarightangle).Theratiothenbecomes3/4.'circumferencelengthofgearAtothatofB?Aratioisalwaysanabstractnumber;i.e.,ithasnounits.Itisanumber-)consideredapartfromthemeasuredunitsfromwhichitcame.ThusinFig.:J36teeth8.1,theratioofthewidthtothelengthis15to24or5:8.Notethisdoesno~-1124"I15"Fig.B.l.n------Exs.7,B.8.IfgearAturns400timesaminute,howmanytimesaminutewillgearBturn?9.Inaschoolthereare2200pupilsand105teachers.Whatisthepupil-teacherratio?Exercises10.Express37percentasaratio.11.Thespecificgravityofasubstanceisdefinedastheratiooftheweightofagivenvolumeofthatsubstancetotheweightofanequalvolumeofwater.1.Expressinlowesttermsthefollowingratios:Ifonegallonofalcoholweighs6.8poundsandonegallonofwater(a)8to12.weighs8.3pounds,whatisthespecificgravityofthealcohol?(b)15to9.12.Onemile=5280feet.Akilometer=3280feet.Whatistheratioofa(c)i~.kilometertoamile?(d)2xto3x.13.Whatistheratioofthelengthofthecircumferenceofacircletothe(e)l~Stot.lengthofitsdiameter?14.Themeasuresoftheacuteanglesofarighttriangleareintheratioof7to2.Whatistheratioof:8.Howlargearethemeasuresoftheangles?(a)1rightLto1straightL?-- 248FUNDAMENTALSOFCOLLEGEGEOMETRY15.Draw6ABC,asinthefigureforEx.15,withmAB=8centimetetjPROPORTION-SIMILARPOLYGONS249mCB=6centimeters,mLC=1rightLandCD..lAB.MeasureThefourthproportionaltothreequantitiesisthefourthtermoftheCD,AD.andBDaccuratelyto~centimeter.Expresstheratio,Alroportion,thefirstthreetermsofwhicharetakeninorder.ThusinthemABImACandmBCImCDtothenearesttenth.Proportiona:b=e:d,disthefourthproportionaltoa,b,ande.PWhenthesecondandthirdtermsofaproportionareequal,eitherissaidtoBcbethemeanproportionalbetweenthefirstandfourthtermsofthepropor-tion.Thus,ifx:y=y:z,yisthemeanproportionalbetweenxandz.Ifthreeormoreratiosareequal,theyaresaidtoformaseriesofequalratios.Thusa/x=b/y=e/zisaseriesofequalratiosandmayalsobewritten6emintheforma:b:e=x:y:z.8.3.Theoremsaboutproportions.Sinceaproportionisanequation,a]]axiomswhichdealwithequalitiescanbeappliedtoaProportion.AlgebraictheA8emAmanipulationofproportionswhichchangetheformoccursofrequentlywhereverproportionsareusedthatitwi]]beusefultolistthemasfo]]ows:Ex.15.Ex.16.Theorem8.1.Inaproportion,theproductoftheextremesisequaltothe16.Draw6ABCwithmLA=50,mAC=8centimeters,mAB=10centimeteproductofthemeans.ThendrawMN1/AB.MeasureAM,MC,BN,andNCtothenearttenthofacentimeter.ExpresstheratiosofmAMImMCandmBN1m.aetothenearesttenth.Given:b=;j'17.Draw00withradius=5centimeters.Conclusion:ad=be.DrawchordsAD=6centimeters,CB=8Acentimetersanyplaceonthecircle.DrawABandCD.MeasureAE,EB,CD,andProofDEaccuratelytoI;)centimeter.ExpresstheratiosofmDEImBEandmAEImCESTATEMENTSREASONStuthenearesttenth.D----aeEx.17.1--.-bd'1.Given.8.2.Proportion.Aproportionisanexpressionofequalityoftworatios.2.bd=bd.2.Reflexiveproperty.example,since6/8and9/12havethesamevalue,theratioscanbeequatedae3.proportion,6/8=9/12or6:8=9:12.Thus,ifratiosa:bande:dareeq.~bXbd=dXbd,orad=be.3.Multiplicationproperty.theexpressiona:b=e:disaproportion.Thisisread,"aistobaseisto:j--or"aandbareproportionaltoeandd".Intheproportion,aisreferre1sRAPEx.15.Ex.16.PTheorem8.15P8.20.Thealtitudeonthehypotenuseofarighttriangleformstworighttriangleswhicharesimilartothegiventriangleandsimilartoeachother.Given:6ABCwithLACBarightL;PJAPPCEx.9.=~.Ex.10.pc-To-HPSHCD.1AB.r~~------ PROPORTION-SIMILARPOLYGONS271270FUNDAMENTALSOFCOLLEGEGEOMETRYCcConclusion:RightL.ADC~rightL.ACB;rightL.CDB~rightL.ACB;rightL.ADC~rightL.CDB.Given:L.ABCwithLACBarightL./1:Conclusion:c2=a2+b2.rSA/1BBDAI.DProofcJTheorem8.15.ProofTheorem8.16.REASONSSTATEMENTSSTATEMENTSREASONS1.LACBisarightL.1.Given.2.Given.1.DrawCD1-AB.1.Theorem5.3;theorem5.4.2.CD1-AB.3.§1.20.2.c:a=a:sandc:b=b:r.2.§8.22.3.LADCandLBDCarerightA.4.Inright&ADCandACB,LA==4.Reflexiveproperty.3.a2=cs,b2=cr.3.Theorem8.1.4.([2+b2=cs+cr.4.E-4.LA.5.:.L.ADC~L.ACB.5.§8.12.5.([2+b2=c(s+r).5.Factoring(distributivelaw).6.Inright&BDCandABC,LB==6.Reflexiveproperty.6.s+r=c.6.§1.13.([2LB.7.+fi=c2.7.E-8.7.:.L.CDB~L.ACB.7.Reason5.8.c2=a2+b2.8.E-2.8.:.L.ADC~L.CDB.8.§8.13.Theorem8.16isknownasthePythagoreantheorem.AlthoughthetruthscofthetheoremwereusedformanyyearsbytheancientEgyptians,thefirst8.21.Corollary:ThealtitudeontheformalproofofthetheoremisattributedtothePythagoreans,amathematicalhypotenuseofarighttriangleisthemeansocietywhichwasfoundedbytheGreekphilosopherPythagoras.Sincethatproportionalbetweenthemeasuresofthetimemanvotherproofsofthisfamotlstheoremhavebeendiscovered.segmentsofthehypotenuse.ASuggestions:L.ADC~L.CDBD8.24.Corollary:ThesquareofthemeasureofthelegofarighttriangleisbyTheorem8.15.ThenequaltothesquareofthemeasureofthehypotenuseminusthesquareoftheCorollary8.21.AD:CD=CD:DB.measureoftheotherleg.IllustrativeExample1:8.22.Corollary:Eitherlegofarighttriangleisthemeanproportional.'.betweenthemeasureofthehypotenuseandthemeasureofthesegmentof!.Tthehypotenusecutoffbythealtitudewhichisadjacenttothatleg.Given:RightL.RSTwithSQ1-tohypotenuseRT;RQ=3andQS=5.FindQT.Suggestions:Dropa1-fromCtoAB;L.BDC~L.BCAbyTheorem8.15.Solution:From§8.21,RQ:QS=QS:QT.Substi-ThenAB:BC=BC:BD.tuting,3:5=5:QT.Therefore,3QT=25;QT=¥'Theorem8.16R8.23.Thesquareofthemeasureofthehypotenuseofarighttriangleisequal)tothesumofthesquaresofthemeasuresofthelegs.IllustrativeExample1.[~ 272FUNDAMENTALSOFCOLLEGEGEOMETRYPROPORTION-SIMILARPOLYGONS273IllustrativeExample2:18-25.Findthelengthofsegmentxineachdiagram.Drawaperpendicularifnecessary.Ej.Find]K.Given:Right6H]KwithhypotenuseHK=17,legH]=TSolution:By§8.24,(jK)2=(HKF-(Hj)2.CSubstituting,(jK)2=(17F-(15)2~~'"~J=289-225iH~~A30BRxS=64.:.]K=8.Ex.18.Ex.19.IllustrativeExample2.ExercisesIn6ABC,LACBisarightL,andCD..LAB.c1.FindCDifAD=9andBD=4.2.FindBCifAB=16andBD=4.C163.FindBCifAD=12,AC=15,andCD=9.4.FindACifAD=24,CD=18,andBC=22.5."u5.FindACifBD=9,BC=15,andCD=12.6.FindCDifAC=20,andBC=15.A/1BD12BH16J7.FindBDifAC=21,andCD=15.Exs.1-12.Ex.20.Ex.21.8.FindACifBD=12,BC=13,andCD=5.9.FindBDifAD=2andCD=4.10.FindCDifAD=16andBD=4.11.FindBCif/lB=20andBD=5.R20sE12.FindACifAB=18andAD=8.13.Given:ABisadiameterof0O,CD..LAB.IfAD==3andBD=27,findCD.AA(Hint:DrawchordsACandBC.)36Ex.22.Ex.23.Ex.13.In6MNT,LMNTisarightL.14.FindMTifMN=16andNT=12.15.FindNTifMN=24andMT=30.M~~Gp16.FindMNifMT=13andNT=5.17.FindNTifMN=15andMT=17.Exs.14-17.Ex.24.Ex.25.(I 274FUNDAMENTALSOFCOLLEGEGEOMETRYPROPORTION-SIMILARPOLYGONS275Theorem8.17Proof8.25.Iftwochordsintersectwithinacircle,STATEMENTSREASONStheproductofthemeasuresofthesegmentsAofonechordisequaltotheproductofthe1.DrawchordsI'AandTB.1.Postulate2.measuresofthesegmentsoftheother.2./IlLTAP=tmfB.2.Theorem7.3.Given:00withchordsABandCDintersect-3./IlLBTP=tmfB.3.Theorem7.14.IingatE.D4.LTAP==LBTP.4.Theorem3.4.IConclusion:AEXEB=CEXED.5.LP==LP.5.Reflexivity.Theorem8.17.6.L,.TAP~L,.BTP.6.§8.11.Proof7.PB:PI'=PI':PA.7.§8.9.STATEMENTSREASONS1.DrawchordsCBandAD.1.Postulate2.Theorem8.192.LDAE==LBCE.2.§7.17.3.LAED==LCEB.3.Theorem3.12.8.28.Iftwosecantsaredrawnfromthesamepointoutsideacircle,the4.:.L,.AED~L,.CEB.4.§8.11.productofthemeasuresofonesecantanditsexternalsegmentisequalto5.AE:CE=ED:EB.5.§8.9.theproductofthemeasuresoftheothersecantanditsexternalsegment.6.:.AEXEB=CEXED.6.Theorem8.1.8.26.Segmentofasecant.WhenacircleiscutbyaGiven:00withsecantsPApsecant,asAPinFig.8.6,andPCdrawnfromP.wespeakofthesegmentAPConclusion:PAXPB=PCXPD.DasasecantfromPto00.PBistheexternalsegmentProofofthesecantandBAistheinternalsegmentoftheFig.8.6.secant.Theorem8.19.Theorem8.18STATEMENTSREASONS8.27.IfatangentandasecantaredrawnI.DrawchordsADandBe.1.Postulate2.fromthesamepointoutsideacircle,the2.mLDAP=trnBD.2.Theorem7.3.measureofthetangentisthemeanpropor-3.mLBCP=trnBD.3.Theorem7.3.tionalbetweenthemeasuresofthesecant4.LDAP==LBCP.4.Theorem3.4.anditsexternalsegment.5.LP==LP.5.Reflexiveproperty.6.L,.DAP~L,.BCP.6.§8.11.Given:00withtangentPI'andsecantPBA7.PA:PC=PD:PB.7.§8.9.drawnfromP.8.PAXPB=PCXPD.8.Theorem8.1.Conclusion:PB:PI'=PI':PA.Theorem8.18.--------------------- 276PROPORTION-SIMILARPOLYGONS277FUNDAMENTALSOFCOLLEGEGEOMETRY...----------..Exercises14.FindPAifPC=24,PB=10,PD=8.15.FindPBifAP=18,PC=24,PD=6.16.FindPCifPD=6,PB=8,BA=10.Inthefollowingexercises,0isthecenterofthecircle.17.FindADifAP=16,BC=12,PC=20.1.FindAEifEB=4,CE=8,ED=5.18.FindPDifPB=8,AD=10,BC=16.19.FindEDifOA=8,OE=3,CE=10.2.FindEDifAE=12,CE=8,EB=6.2().FindBDifOA=8,CD=3,AD=5.3.FindCEifAB=20,EB=15,ED=7.2!.FindOAifAD=8,BD=5,CD=4.C4.FindACifCE=9,EB=3,BD=5.Exs.I-5.Exs.20,21.5.FindCDifEB=6,AB=18,ED=8.6.FindPTifPS=4,PR=9.7.FindPRifPS=5,PT=8.8.FindPTifRS=7,PR=16.9.FindRTifPT=18,1'S=9,PS=12.10.FindPTifRS=24,PS=8.n11.FindPi'ifOA=15,P/l=10.12.FindAPifPT=12,OA=9.13.FindOAifPT=8,PA=4.TExs.ll-I3.pABExs.14-18.Ex.19.------------------ Test2TRUE-FALSESTATEMENTS1.Aproportionhasfourunequalterms.2.Iftwotriangleshavetheircorrespondingsidescongruent,thentheircorrespondinganglesarecongruent.3.Iftwotriangleshavetheircorrespondinganglescongruent,thentheircorrespondingsidesarecongruent.4.Themeanproportionalbetweentwoquantitiescanbefoundbytakingthesquarerootoftheirproduct.5.Twoisoscelestrianglesaresimilarifanangleofoneiscongruenttoacorrespondingangleoftheother.SummaryTests6.Thealtitudeonthehypotenuseofarighttriangleisthemeanpropor-tionalbetweensegmentsofthehypotenusecutoffbythealtitude.7.Oftwounequalchordsofthesamecircle,thegreaterchordisthefartherfromthecenter.8.Ifalinedividestwosidesofatriangleproportionately,itisparalleltotheTest1thirdside.9.Iftwopolygonshavetheircorrespondingsidesproportional,theyareCOMPLETIONSTATEMENTSsimilar.I.Givenright/:::.MNPwithLNarightangleandNTthealtitudeonMP.10.Twoisoscelesrighttrianglesaresimilar.ThenNPisthemeanproportionalbetweenandMP.11.Thesquareofthehypotenuseofarighttriangleisequaltothesumofthe2.Astatementofequalityoftworatiosistermedalegs.3.Iftwopolygonshavethesameshape,theyare-12.Ifalinedividestwosidesofatriangleproportionately,itisequaltohalf4.IfchordsAINandRSof00intersectatP,thenRP:MP=thethirdside.5.IfABCandEi5CaresecantsfromexternalpointCto00,thenACXBC13.Thediagonalsofatrapezoidbisecteachother.X.14.Iftwotriangleshavetwoanglesofonecongruentrespectivelytotwo6.IffiRisatangentandfiTSisasecantof00drawnfromexternalpointanglesoftheother,thetrianglesaresimilar.PthroughpointsTandSofthe0,thenPSXPT=X15.Correspondingaltitudesofsimilartriangleshavethesameratioasthatof7.Theperimeterofarhombushavingdiagonalsof6inchesand8inchesisanytwocorrespondingsides.16.Congruentpolygonsaresimilar.inches.17.Iftwochordsintersectwithinacircle,thesumofrhesegmentsofone8.Thesquareofalegofarighttriangleequalsthesquareofthehypotenusethesquareoftheotherleg.chordequalsthesumofthesegmentsoftheother.18.Ifatangentandasecantaredrawnfromthesamepointoutsideacircle,xax+y9.If-y=~then-y.,=thetangentisequaltoone-halfthedifferenceofthesecantanditsexternalb'segment.10.Ifxy=1'5,thenx:s=19.IftworighttriangleshaveanacuteangleofonecongruenttoanacuteII.Themeanproportionalbetween4and9isangleoftheother,thetrianglesarecongruent.12.Thefourthproportionalto6,8,12is20.Twotrianglescongruenttothesametrianglearesimilartoeachother.13.If7a=3b,thena:b=14.b:5=a:1O~a:b=Test315.8:x=5:y~x:y=PROBLEMS16.ay=bx~x:y=278279f~ 280FUNDAMENTALSOFCOLLEGEGEOMETRYSUMMARYTESTS281Findthevalueofxineachofthefollowing:EcT105m~cy~15A6~ABRS 9Prob.i.DEIIAB.Prob.2.PQIIRS.DProb.9.IIIInIIn.Prob.10.ED=20;BD=x.DProb.3.Prob.4.xBProb.ii.BDbisectsLADC;AD24;=Prob.i2.DC=25;DE=20;AE=16;BC=x.xProb.5.Prob.6.2320~1812--JProb.7.Prob.8.Prob.i3.Prob.i4.F=='='l-----------------------------------..-------------~- est4191XERCISESuPTGiven:0URSTwithdiagonalRT.Prove:PM:SM=MT:RM.[/(JRSEx.i.LInequalities2.Given:HK=LK;IKbisectsLHKJ.~Prove:LH:HI=KI:IJ.9.1.Inequalitiesarecommonplaceandimportant.InourstudythusfarweHIJhavefoundvariouswaysofprovingthingsequal.Oftenitisequallyimport-Ex.2.anttoknowwhenthingsareunequal.Inthischapterwestudyrelationshipsbetweenunequallinesegments,angles,andarcs.E9.2.Orderrelations.SincewewillusetheorderrelationsgiveninChapter3asoperatingprinciplesinthischapter,thestudentisadvisedtoreviewthemat3.Given:AOBadiameterof00;thistime.ThestudentwillnotetherelationbetweenPostulates13and14DE1-AOBextended.andthepartitionproperty(0-8).InFig.9.1,weusePostulate13tojustifytheProve:L.ADE~L.ACB.ArelationshipmAB+mBC=mAC(orAB+BC=AC).Postulate14canbeDcitedtoexpresstherelationmLABC=mLABD+mLDBC.Byusingthepartitionproperty,itimmediatelyfollowsthat1.mAC>mABandmAC>mBG.Ex.3.2.mLABC>mLABDandmLABC>mLDBC.OftenABandBCarereferredtoasthepartsofAC,whileLABDandLDBCarethepartsofLABC.Thusthepartitionpropertycouldbestatedas:"Thewholeisgreaterthanitsparts."~1~IPORTANT.Hereafterwewillfrequentlyfollowthepracticeofrefer-ringtoagivensegmentasbeing"equalto"or"greaterthan"anotherseg-mentinsteadofstatingthatthe"measureofonesegmentisequalto"or"themeasureofonesegmentisgreaterthan"themeasureofasecondseg-ment.ThestudentisremindedthatwearenowusingmABandABinter-changeably,andwearenowacceptingmAB=mCD,AB~CD,andAB=CDasequivalentstatements.Also"AB>CD"willbeconsideredequivalent282283[- 284FUNDAMENTALSOFCOLLEGEGEOMETRYINEQUALITIES285tomAB>mCDandAB+BCandmAB+mBCwillbetwowaysusedtosay;Exercises(A)thesamething.ThispracticewillbefollowedinordertoshortenotherwiselongandAnswereachquestion.Ifnoanswerispossible,indicatewith"noanswerpossible."1.BillhasmoremoneythanTom.Eachearnedanadditional10dollars.HowdoBill'sandTom'stotalamountscompare?2.BillhasmoremoneythanTomandFrankhaslessthanJohn.HowdoBill'sandJohn'scompare?3.BillhasthesameamountofmoneyasAlice.Alicespendsmorethan...Bill.Howthendotheirremainingamountscompare?ABCBA4.JohnhasmoremoneythanTom.Johnloseshalfhismoney.Howdo(a)(b)theirremainingamountscompare?Fig,9.1.5.BillhaslessmoneythanMary.Eachdecidestogivehalfofhismoneytocharity.Howdotheamountstheyhaveleftcompare?cumbersomestatements.Thestudentshouldkeepinmind,however,that-6.JohnhasmoremoneythanTom.Eachdoubleshisamount.Who,themeasuresofgeometricfiguresarebeingcomparedintheseinstances.~-then,hasthemoremoney?'7.AnnisolderthanAlice.MaryisyoungerthanAlice.CompareMary's'9.3.Senseofinequalities.Twoinequalitiesareofthesamesenseifthesame'andAnn'sages.':symbolisusedintheinequalities.Thusadare>9.AnnandBillareofdifferentages.MaryandTomarealsoofdifferentofoppositesense..'ages.CompoTetheagesofAnnandTom.Astudyofthebasicpropertiesofandtheoremsforinequalitieswillrevearl~10.JohnhastwiceasmuchmOTleY;j<;MiI'Y,,mdMaryprocesseswhichwilltransformanInequalItytoanotherInequalItyofthesame~asAlice.ComparetheamountsofJohnandAlice.sense.Someofthemare:'(a)Addingequalrealnumberstobothsidesofaninequality.)(b)Subt,",cin~equaltcalnumbmhomboth,(de>ofaninequality.l,I(c)Multiplyingbothsidesofaninequalitybyequalpositiverealnumbers.Exercises(B)(d)Dividingbothsidesofaninequalitybyequalpositiverealnumbers.(e)Substitutinganumberforitsequalinaninequality.Copyandcompletethefollowingexercises.Thefollowingprocesseswilltransformaninequalitytoanotherinequality,Ifnoconclusionispossible,writeaquestionmarkinplaceoftheblank.ofoppositesense:(a)Dividingthesame(orequivalent)positivenumberbyaninequality.II.Ifa>bandc=d,thena+c(b)Subtractingbothsidesofaninequalityfromthesamerealnumber.b+d.12.Ifrb,thenr-b.(c)Multiplyingbothsidesofaninequalitybythesamenegativenumber.s-x.13.Ifx=2y,r=2s,andykandk>m,thenlm.[Note:Todividebyanumberaisthesameastomultiplybyitsmultiplicativ>:15.Ifx+y=z,thenzx.inversel/a.]i,16.IfAB+BC>AC,thenABAC-BC.----------------------------[.. INEQUALITIES287286FUNDAMENTALSOFCOLLEGEGEOMETRYc17.IfLx==LYc~.YH/23.IfLaisanexteriorLofandmLr>mLs,then:y6ABC,thenmLamLA.mLABCmLDEF.~B~E~A.,.Ex.23.BDEx.17.24.!fmLR>mLS,18.IfBC.1AB;EF.1DE;c~FLRKbisectsLSRT,andmLf3BEandEC>CD,thenACltL.Given:6ABCwithBC>AC.Conclusion:mLBAC>mLB.ABProofTheorem9.1.ABExs.19,20.ISTATEMENTSREASONScI.BC>AC.I.Given.2.OnCBletDbeapointsuchthat2.PostulateII.CD=AC.21.IfmLCAB=mLABC,3.DrawAD.thenBDAC.3.Postulate2.4.mLa22.!fAC=BC,=mLf3.4.Theorem4.16.5.mLBACthenmLABCmLBAD.=mLBAD+mLa.5.Postulate14.6.mLBAC>mLa.6.0-8.7.mLBAC>mLf3.7.0-7.8.mLf3>mLB.8.Theorem4.17.AB9.--".mLBAC>mLB.9.0-6.Exs.21,22.!=------.-----l- INEQUALITIES289288FUNDAMENTALSOFCOLLEGEGEOMETRY9.8.ThesumofthemeasuresoftwosidesofatriangleisgreaterthantheTheorem9.2measureofthethirdside.9.5.Iftwoanglesofatrianglearenotcongruent,thesideoppositethelargerGiven:L.ABC.ofthetwoanglesisgreaterthanthesideoppositethesmallerofthetwoConclusion:AB+BC>AC.angles.ProofCSTATEMENTSREASONSGiven:L.ABCwithmLA>mLB.I.LetDbethepointontherayI.PostulateII.Conclusion:BC>AC.oppositeifCsuchthatDB=AB.A~B2.DrawAD.2.Postulate2.Theorem9.2.3.DC=DB+BC.3.Postulate13.Proof4.DC=AB+BC.4.£-8.REASONS5.mLDAC=mLDAB+mLBAC.5.Postulate14.STATEMENTS6.mLDAC>mLDAB.6.0-8.1.Given.7.mLDAB=mLADB.7.Theorem4.16.I.mLA>mLB.2.Trichotomyproperty.8.mLDAC>mLADB.8.0-7.2.InL.ABC,sinceBCandACare9.DC>AC.9.Theorem9.2.realnumbers,thereareonlytheIO.AB+BC>AC.10.Substitutionproperty.followingpossibilities:BC=AC,BCAC.3.AssumeBC=AC.Thistheoremmaybeusedtoshowthattheshortestroutebetweentwo4.ThenmLA=mLB.pointsisthestraightlineroute.5.Statement4isfalse.Theorem9.46.NextassumeBC<.1.c.7.ThenmLAAC.ofthefirstisgreaterthanthethirdsideofthesecond.9.6.Corollary:TheshortestsegmentjoiningapointtoalineFperpendicularsegment.cNote.Herewecanprovewhatwestatedin§1.20.9.7.Corollary:ThemeasureoftheDA~RD~EABhypotenuseofarighttriangleisgreater'---'........thanthemeasureofeitherleg.I-......BI.:Theorem9.3..:~I1.(!.Theorem9.4.Theorem9.3.Ar------------------ INEQUALITIES291290FUNDAMENTALSOFCOLLEGEGEOMETRY9.11.IllustrativeExample1:Given:LABCandLDEFwithAC=DF,CB=FE,andmLC>mLF.Given:DapointintheinteriorofLABC;AC=CD.Conclusion:AB>DE.Prove:DBAG.15.Theorem9.3.16.BH+AH>AG.16.0-7.Proof17.BH+AH=AB.17.Postulate13.18.AB>AG.18.0-7.19.AB>DE.19.0-7.RsillustrativeExample2.Theorem9.5STATEMENTSREASONS9.10.IftwotriangleshavetwosidesofonecongruentrespectivelytotWOI.S1'=RT.1.Given.sidesoftheotherandthethirdsideofthefirstgreaterthanthethirdsideof-~2.mLS=mLR.2.4.16.thesecond,themeasureoftheangleoppositethethirdsideofthefirstis_,3.mLRK1'>mLS.3.4.17.greaterthanthemeasureoftheangleoppositethethirdsideofthesecond..'4.mLRK1'>mLR.4.0-7.5.R1'>KT.5.Theorem9.2.(Note:Thistheoremisprovedbytheindirectmethod.TheproofislefttOi!6.S1'>KT.6.0-7.thestudent.)--r- 292FUNDAMENTALSOFCOLLEGEGEOMETRYINEQUALITIES293Exercises7.Given:DC=Be.DProve:mLADC>mLA.I.InLABC,mLA=60,mLB=70.Whichis(a)thelongestside;(b)the8.Given:DC=BC.shortestside?,IProve:AD>BD.pc2.Isitpossibletoconstructtriangleswithsidesthelengthsofwhichare:9.Given:DC=Be.(a)6,8,10;(b)1,2,3;(c)6,7,8;(d)7,5,I.Prove:mLCDB>mLA.10.Given:DC=Be.ACProve:AC>De.Exs.7-1O.3.Given:AC=Be.pProve:AC>DC.~BAD11.Given:RP=RS;Ex.3.PT=ST;RTPT>RP.CProve:mLPRS>mLPTS.4.Given:BC>AC;ADbisectsLBAC;sBDbisectsLABC.Ex.n.Prove:BD>AD.A6BII12.Given:AMisamedianofLABe.cEx.4.Pmv"AMAB.'s','~,Prove:DC>DB.~,,.'.BABc,:::,:,::,.._,,~,Ex.5.'13.ItisdesiredinthefiguretofindtheshortestpathfrompointAtolineATandthentopointB.ProvethattheshortestlineisthebrokenlinexformedwhichmakesLex~L{3.Given:Lex~L{3.Prove:AR+RBQR.RsEx.6.Ex.l3. 294FUNDAMENTALSOFCOLLEGEGEOMETRYINEQUALITIES295ProofSTATEMENTSREASONSSTATEMENTSREASONS1.00==OQ.1.Given.2.mLO>mLQ.2.Given.I.DrawBM..lA.I.Why?3.mLO=mAR,mLQ=mCD.3.§7.9.2.ExtendBMandARuntilthey2.Why?4.:.mAR>mCD.4.0-7.intersectatS.DrawTS.3.La==Lf3.3.Why?4.La==Ly.4.Why?Theorem9.75.L?==L?5.Why?6.RM=RM.6.Why?9.14.Inacircleorincongruentcircles,iftwominorarcsarenotcongruent,7.LRMB==LRMS.7.Why?thegreaterarchasthegreatercentralangle.8.RB=RS.8.Why?(Theproofofthistheoremislefttothestudent.)9.BM=SM.9.Why?10.~=~.10.Why?Theorem9.8II.LIMB==LTMS.II.Why?9.15.Inacircleorincongruentcircles,thegreateroftwononcongruent12.TB=TS.12.Why?chordshasthegreaterminorarc.13.ASchordCD.Conclusion:mAR>mCD.QProofSTATEMENTSREASONS1.00==OQ.I.Given.Theorem9.6.2.DrawradiiOA,OB,QC,QD.2.Postulate2.3.OA=QC;OB=QD.3.Definitionof@.==4.ChordAB>chordCD.4.Given.Given:00==OQwithmLO>mLQ.5.mLO>mLQ.5.Theorem9.5.Conclusion:mAR>mCD.6.mAR>mCD.6.Theorem9.6.Proof 296FUNDAMENTALSOFCOLLEGEGEOMETRYINEQUALITIES297ExercisesTheorem9.9Ineachofthefollowingcircles,0isassumedtobethecenterofacircle.9.16.Inacircleorincongruentcirclesthegreateroftwononcongruentminorarcshasthegreaterchord.1.In,0.ABC,mAB=4inches,mBC=5inches,mAC=6inches.Namethe(Theproofofthistheoremislefttothestudent.)anglesofthetriangleinorderofsize.c2.,0.RSTisinscribedinacircle.mRS=80andmST=120.NametheanglesTheorem9.10ofthetriangleinorderofsize.3.In,0.MNT,mLN=60andmLMmLN.Whichisthegruent,theyareunequallydistantlonger,NTorLT?Proveyouranswer.fromthecenter,thegreaterchord5.InquadrilateralQRST,QR>RSandLQ==LS.Whichisthelonger,beingnearerthecenter.QTorST?Proveyouranswer.Given:00withchordAB>chordCD;cOE1-AB;OF1-CD.Conclusion:OEAB.REASONS'STATEMENTSProve:AC>AD..':1.DrawchordAH==chordCD.1.Postulate11.1;"2.DrawOG1-AH.2.Theorem5.4..I3.DrawGE.3.Postulate2..'cEx.6.4.OE1-AB.OF1-CD.4.Given.1iA---7-6.AB>CD.6.Given.7.Provethat,inacircle,ifthemeasureofoneminorarcistwicethemeasure7.AB>AH.7.0-7.£-1.ofasecondminorarc,themeasureofthechordofthefirstarcislessthan8.GismidpointofAH;Eismid-8.Theorem7.7.twicethemeasureofthechordofthesecondarc.pointofAB.8.Provethat,ifasquareandanequilateraltriangleareinscribedinacircle,9.AE>AG.9.0-5.thedistancefromthecenterofthecircletothesideofthesquareis10.mLOI>mL{3.10.Theorem9.1.greaterthanthattothesideofthetriangle.11.mLAGO=mLAEO.11.§1.20;Theorem3.7.12.mLy.12.0-3.13.OEOA.circletothecircleisalongaProve:mLPOR>mLROS.radiusproduced.(Hint:11.Given:00withOA1-PR;PS+SO>...;SO=RO;OB..lSR;PS>....)mLPOR>mLROS.Ex.19.Prove:OB>OA.Exs.10,11.R12.Given:00withPT=ST;20.Given:ChordAC>chordBD.mLSTR>mLRTP.Prove:ChordAB>chordCD.sProve:mRS>mFR.13.Given:00withPT=ST;mRS>mPR.Ex.20.Prove:mLSTR>mLRTP.Exs.12,13.14.Provethatthemeasureofthehypotenuseofarighttriangleisgreaterthanthemeasureofeitherleg.15.Provethattheshortestchordthroughapointinsideacircleisperpendi-culartotheradiusthroughthepoint.16.Given:CMisamedianof,6.ABC;CMisnot1-AB.Prove:AC#-BC.17.Given:CMisamedianof,6.ABC;Exs.16,17.AC#-Be.Prove:CMcannotbe..lAB.18.Provethattheshortestdis-tancefromapointwithinacircletothecircleisalongaradius.(Hint:ProvePBmLT,thenisthelongestsideof-thetriangle.20.Theshortestchordthroughapointinsideacircleistotheradius-throughthepoint.Test2TRUE-FALSESTATEMENTSI.Theshortestdistancefromapointtoacircleisalongthelinejoiningthatpointandthecenterofthecircle.2.Themeasureoftheperpendicularsegmentfromapointtoalineistheshortestdistancefromthepointtotheline.SummaryTests3.Eitherlegofarighttriangleisshorterthanthehypotenuse.4.Iftwotriangleshavetwosidesofoneequaltotwosidesoftheother,andthethirdsideofthefirstlessthanthethirdsideofthesecond,themeasureoftheangleincludedbythetwosidesofthefirsttriangleisgreaterthanthemeasureoftheangleincludedbythetwosidesoftheTest1second.5.:-../0twoanglesofascalenetrianglecanhavethesamemeasure.COMPLETIONSTATEMENTS6.Themeasureofanexteriorangleofatriangleisgreaterthanthemeasurethe;:ofany'wo,id"ofa"ianglei,-thanIofanyoftheinteriorangles.I.Thewmofthemeaw'"measureofthethirdside..7.Iftwosidesofatriangleareunequal,themeasureoftheangleopposite'2.AngleTisthelargestangleintriangleRST.Thelargestsideis-'...:thegreatersideislessthanthemeasureoftheangleoppositethesmaller'3.Ifk>h,thenk+l_h+1.,cside..,1.:...m<.:17andn<:p,..13-I[:L,'.'(chnrdsinthesamecircleareunequal,thesmallerchordisnearerLltCHrnp.5.IfI>w,thena-l_a-w.thecenter.9.IfJohnisolderthanMary,andAliceisyoungerthanMary,Johnisolder6.IfdJK,mLJ=80.ThenmLH_50.thanAlice.10.BillhastwiceasmuchmoneyasTom,andTomhasone-thirdasmuchas8.InquadrilateralLMNP,LM=MNandmLMLP>mLMNP.mLNLPmLLNP.Harry.ThenBillhasmoremoneythanHarry.-II.AngleQisthelargestanglein6.PQR.ThenthelargestsideisPQ.9.In6.ABC,mLA=50,mLB=60,mLC=70.ThenAB_AG.12.Ifk>mandmt.10.Inacircleorincongruentcircles,iftwocentralanglesarenotcongruent,'.:13.lfx>0andy>0,thenxyd,thena+d_b+c.Y>x.17.Thedifferencebetweenthelengthsoftwosidesofatriangleislessthan14.Ifxy,thenz_x.IS.Theperimeterofaquadrilateralislessthanthesumofitsdiagonals.16.Ifxy0,theny_O..19.Ifatriangleisnotisosceles,thenamediantoanysideisgreaterthanthe17.InquadrilateralPQRS,ifPQ=QR,andmLP>mLR,thenPS_RS_RS,altitudetothatside.18.InquadrilateralPQRS,ifPQ>QR,andmLP=mLR,thenPS300301 20.Thediagonalsofarhombusthatisnotasquareareunequal.1101Test3EXERCISEScI.Supplythereasonsforthestatementsinthefollowingproof:Given:LABCwithCDbisectingLACB./TProve:AC>AD.GeometricConstructionsADBProofEx.I.STATEMENTSREASONS10.1.Drawingandconstructing.Inthepreviouschaptersyouhavebeen1.mLACD=mLBCD.1.drawinglineswitharulerandmeasuringangleswiththeprotractor.Mathe-2.mLADC>mLBCD;2.maticiansmakeadistinctionbetweendrawingandconstructinggeometricmLBDC>mLACD.figures.Manyinstrumentsareusedindrawing.Thedesignengineerand3.mLADC>mLACD.3.thedraftsmanindrawingblueprintsforairplanes,automobiles,machine4.AC>AD.4.parts,andbuildingsuserulers,compasses,T-squares,parallelrulers,anddraftingmachines.Anyorallofthesecanbeusedindrawinggeometricfigures(Fig.10.1).pWhenconstructionsare.made,theonlyinstrumentspermittedaretheMFaightcdgc(anunmarkedruin)andacompass.TIle~11di~IILed~eisused2.Given:PR=PT.forconstructingstraightlinesandthecompassisusedfordrawingcirclesorProve:mLPRS>mLS.arcsofcircles.Itisimportantthatthestudentdistinguishbetweendrawingandconstructing.Whenthestudentistoldtoconstructafigure,hemustRnotmeasurethesizeofangleswithaprotractororthelengthoflineswitharuler.Hemayuseonlythecompassandthestraightedge.Ex.2.Ifwearetoldtoconstructthebisectorofanangle,themethodusedmustbesuchthatwecanprovethatthefigurewehavemadebisectsthegivenangle.3.Provethattheshortestchord10.2.Whyuseonlycompassandstraightedge?TherestrictiontotheuseofonlyacompassandstraightedgeonthegeometrystudentwasfirstestablishedthroughapointwithinacircleiscbytheGreeks.Itwasmotivatedbytheirdesiretokeepgeometrysimpleandperpendiculartotheradiusdrawnthroughthatpoint.aestheticallyappealing.TothemtheintroductionofadditionalinstrumentsEx.J.wouldhavedestroyedthevalueofgeometryasanintellectualexercise.This~ntroductionwasconsideredunworthyofathinker.TheGreekswerenotInterestedinthepracticalapplicationsoftheirconstructions.Theywere303302---------------------------------l 304FUNDAMENTALSOFCOLLEGEGEOMETRYGEOMETRICCONSTRUCTIONS305rABFig.10.2.3.Twocoplanarnonparallellinesintersectinapoint(Theorem3.1).4.Twocircles0andPwithradiiaandbintersectinexactlytwopointsifthedistancecbetweentheircentersislessthanthesumoftheirradiibutFig.10.1.greaterthanthedifferenceoftheirradii.Theintersectionpointswilllieindifferenthalf-planesformedbythelineofcenters(Fig.10.3).fascinatedbyexploringthemanyconstructionspossiblewiththeuseofonly5.Alineandacircleintersectinexactlytwopointsifthelinecontainsatheinstrumentstowhichtheyhadlimitedthemselves.pointinsidethecircle.Theconstructionswhichwewillconsiderinthischaptershouldserve:-.--objectivessimilartothosesetbytheearlyGreeks.We,too,willrestrict'abourselvestotheuseofonlythestraightedgeandthecompass.10.3.Solutionofaconstructionproblem.Everyconstructionproblemcanbesolvedbystepsasfollows:a~~pStepI:AstatementoftheproblemwhichteLL,whatistobecoilstructed.bStepII:Afigurerepresentingthegivenparts.StepIII:AstatementofwhatisgivenintherepresentationofStepII.IcStepIV:Astatementofwhatistobeconstructed,thatis,theultimateresulttobe'1I.obtained.Fig.10.3.a+b>c.aStepV:Theconstruction,withadescriPtionofeachstep.AnauthG-bQ2,Q:J,...,Q,,-tdivideABintoncongruentseg-STATEMENTSREASONSments.1.BisecttwoanglesofLoABC.1.Construction2.2.Let0bethepointofintersection2.Theorem3.1.Proofofthebisectors.REASONS3.ConstructODfrom0perpen-3.Construction5.STATEMENTSdiculartoAC.1.Byconstruction.4.With0ascenterandmODas4.Postulate19.1.APt=PtP2=P2P3=...=Pn-IP",2.Byconstruction.asradius,construct0O.2.PIQtIIP2~IIP3QJII...IIPI/B.3.Theorem6.8.:'J.00istheinscribedO.3.AQI=Qt~=Q.2~='"=Q"-IB.(Theproofofthisconstructionisleftasanexercise.)Construction8B_-....Exercises(B)",10.12.Circumscribeacircleaboutatriangle.\1.Drawtwopoints3inchesapart.Locatebyconstructionallthepoints2incheshorneachofthesegivenpoints.Given:LoABC.I2.Drawtwolinesintersectingat45°andtwootherparallellineswhichareToconstruct:Circumscribea0aboutLoABC.I]inchapart.LocatebyconstructionallthepointsequidistantfromtheIintersectinglinesandequidistantfromtheparallellines.Construction:A',I'C:3.Drawtwolines/1and/2intersectingata60°angle.LocatebyconstructionComtruction8..~/II/allthepointsthatare1inchfromIIand12,"---"""4.Drawacircle0witharadiuslengthof2inches.DrawadiameterAB.SLTDIE,;TSRF"SONSLucalebycunstructionthepuintsthatareIinchfromthediameterABandequidistanthornAandO.1.Constructtheperpendicularbi-1.Construction4.:'J.DrawatriangleABCwithmeasuressectorsoftwosidesoftheLo.ofthesidesequalto2inches,c2.ThetwolinesmeetatapointO.2.Theorem3.1.2tinches,and3inches.Locate3.With()asthecenterandmOA3.Postulate19.byconstructionthepointsontheastheradiusconstructthecircleO.altitudefromBthatareequi-()4.0isthecircumscribedcircle.distantfromBandC.AQ6.InthetriangleofEx.5,locateby3B(Theproofofthisconstructionisleftasanexercise.)constructionallthepointsontheBaltitudefromCthatareequi-Ex.'.5-8.Construction9distantfromsidesABandBC.7.InthetriangleofEx.5,locatebyconstructionallthepointsonthemedian10.13.ToinscribeacircleinagivenfromCthatareequidistantfromAandC.triangle.~.InthetriangleofEx.5,locatebyconstructionallthepointsequidistantGiven:LoABC.fromsidesACandBCatadistanceoftinchfromsideAB.Toconstruct:InscribeacircleinLoABC.9.DrawatriangleABC.LocatebyconstructionthepointPthatisequi-Adistantfromtheverticesofthetriangle.WithPascenterandmPBasConstruction9. radius,constructacircle.(Thiscircleissaidtobecircumscribedaboutthetriangle.)10.DrawatriangleABC.LocatebyconstructionthepointPequidistantfromthesides2Ithetriangle.ConstructthesegmentPMfromPper-pendiculartoAB.WithPascenterandmPMasradius,constructacircle.(Thiscircleissaidtobeinscribedinthetriangle.)SummaryTestConstructionsTest1-4.WithrulerandprotractordrawAB=3inchesandLawhosemeasureis40.].ConstructanisoscelestrianglewithbaseequalingABandbaseanglewithmeasureequalingmLa.2.ConstructanisoscelestrianglewithlegequalingABandvertexanglewithmeasureequalingmLa.j.ConstructanisoscelestrianglewithaltitudetothebaseequalingABandvertexanglewithmeasureequalingmLa.4.ConstructanisoscelestrianglewithbaseequalingABandvertexanglewithmeasureequalingmLa.5-9.WithrulerandprotractordrawsegmentsAB=2inches,CD=3inches,EF=4inches,andLawhosemeasureis40.5.ConstructDPQRSwithPS=AB,PQ=CDandPR=EF.6.ConstructDPQRSwithPS=AB,PR=EF,andSQ=CD.7.ConstructDPQRSwithPS=AB,PR=CD,andLSPR8.ConstructDPQRSwithPQ==La.=AB,PR=EF,andaltitudeonPQ=CD.~J.ConstructDPQRSwithPQ=CD,LSPQ]0.==La,andaltitudeonPQ=AB.Constructananglewhosemeasureis75.]I.Drawaline5incheslong.Thendivideitintofivecongruentsegmentsusingonlyacompassandstraightedge.316317 12.Drawanyf:..ABC.LetPbeapointoutsidethetriangle.FromPcon-structperpendicularstothethreesidesoff:..ABC.Ill/13.Drawanobtusetriangle.Thenconstructacirclewhichcircumscribesthe'~triangle.:i14.Dra~atriangle.Thenconstructacirclewhichisinscribedinthetriangle.~115.Drawanobtusetriangle.Thenconstructthethreealtitudesofthetriangle.16.Drawatriangle.Thenconstructthethreemediansofthetriangle.GeometricLoci..11.1.Lociandsets.Thesetofallpointsisspace.Ageometricfigureisasetofpointsgovernedbyoneormorelimitinggeometricconditions.Thus,a11geometricfigureisasubsetofspace.InChapter7wedefinedacircleasasetofpointslyinginaplanewhichareequidistantfromafixedpointoftheplane.Mathematicianssometimesusetheterm"locus"todescribeageometricIIfigure.-Definition:Alocusofpointsisthesetofallthepoints,andonlythosepoints,whichsatisfyoneormoregivenconditions.Thus,insteadofusingthewords"thesetofpointsPsuchthat...,"wecouldsay"thelocusofpointsPsuchthat...."Acirclecanbedefinedasthe"locusofpointslyinginaplaneatagivendistancefromafixedpointoftheplane."Sometimesonewillfindthelocusdefinedasthepathofapointmovingaccordingtosomegivenconditionorsetofconditions.Considerthepathofthehubofawheelthatmovesalongalevelroad(Fig.11.1).A,B,C,Drepresentpositionsofthecenterofthewheelatdifferentinstantsduringthemotionofthewheel.Itshouldbeevidenttothereaderthat,asthewheelrollsalongtheroad,thesetofpointswhichre-presentthepositionsofthecenterofthehubareelementsofalineparalleltotheroadandatadistancefromtheroadequaltotheradiusofthewheel.Wespeakofthislineas"thelocusofthecenterofthehubofthewheelasthewheelmovesalongthetrack."Inthistextlocuslineswillbedrawnwith318319~. GEOMETRICLOCI321320FUNDAMENTALSOFCOLLEGEGEOMETRYStepIV:Proveyourconclusionbyprovingthatthefiguremeetsthetwocharacteris-ticsoflocilistedin§11.1.Oneofthedifficultiesencounteredbythestudentofgeometryisthatofdescribingthegeometricfigurewhichrepresentsthelocus.Thesedescrip--8f?faA-tionsmustbepreciseandaccurate.~~Il.3.IllustrativeExample1.What/fR2Fig.11.1.isthelocusofthecenterofacirclewithradiusR2thatrollsaroundtheoutsideofasecondcircletheradiusfromgivenandconstructionlongdashlinestodistinguishthemQIofwhichisRI?Asasecondsimpleillustrationofa--Conclusion:ThelocusisacircleQ2/~locus,considertheproblemoffinding/~4thecenterofwhichisthesameasthatthelocusofpointsinaplane2inchesIofthesecondcircleandtheradiusfromagivenpoint0(Fig.11.2).measureofwhichequalsthesumofLetusfirstlocateseveralpoints,suchPit.0~QthemeasuresofradiiRtandR2.asPt,P2,P3,P4,...,whichare2I4inchesfromO.Obviouslythereareaninfinitenumberofsuchpoints.h3ExercisesIllustrativeExample1,Nextdrawasmoothcurvethrough~//thesepoints.Inthiscaseitappears'--.P2-Q3Usingthemethodoutlinedin§11.2,describethelocusforeachofthethatthelocusisacirclewiththecen-followingexercises.Noproofisrequired.Consideronly2-dimensionalterat()andaradiuswhosemeasureFig.11.2.geometryintheseexercises.is2inches.Q;j,I.ThelocusofpointsequidistantfromtwoparallellinesItand12,Ifnow,conversely,weselectpointssuchasQt,Q2'Q4'...,eachom"2.ThelocusofpointswhichareiinchfromafixedpointP.ofwhichmeetstherequirementofbeing2inchesfrom0,itisevidentthai'3,Thelocusofpoints~inchfromagivenstraightlineI.mecircle.4.ThelocusofthemIdpomtsoftheradIiofagIvenCircleO.Thus,toprovethatalineisalocus,itisnecessarytoprovethefollowmsll.i5.ThelocusofpointsequidistantfromsidesBAandBCofLABC.twocharacteristics:''6.ThelocusofpointsequidistantfromtwofixedpointsAandB.I.Anypointonthelinesatisfiesthegivenconditionorsetofconditions.'.',:'7.Thelocusofpointsoneinchfromacirclewithcenterat0andradius2.fjthu(a)anypaintthatwNlji"th,ginn.wnditian"',,'of",nditia'""""~.Iequalto4inches.8.Thelocusofpointslessthan3inchesfromafixedpointP.thelineor(b)anypointnotonthelinedoesnotsatisfythecondition.'~9.Thelocusofthecenterofamarbleasitrollsonaplanesurface.Thewordlocus(pluralloci,pronounced"10'-si")istheLatinwordmea~in(10.ThelocusofpointsequidistantfromtwointersectingstraightlinesIt,"place"or"location."Alocusmayconsistofoneormorepoints,hnes,;'and12,surfaces,orcombinationsofthese.II.Thelocusofthemidpointsofchordsparalleltoadiameterofagivencircle.11.2.Determiningalocus.LetususetheexampleofFig.12.Thelocusofthemidpointsofallchordswithagivenmeasureofagiventhegeneralmethodofdeterminingalocus.circle.13.Thelocusofthepointsequidistantfromtheendsofa3-inchchordStepI:Locateseveralpointswhichsatisfythegivenconditions.drawninacirclewithcenterat0andaradiusmeasureof2inches.'':StepII:Drawasmoothlineorlines(straightorcurved)throughthesepoints.MI'14.Thelocusofthecentersofcirclesthataretangenttoagivenlineata,,StepIII:Formaconclusionastothelocus,anddescribeaccuratelythegeome.]".,givenpoint.figurewhichrepresentsyourconclusion."... GEOMETRICLOCI323322FUNDAMENTALSOFCOLLEGEGEOMETRYIj15.ThelocusofthevertexofarighttrianglewithagivenfixedhypotenusejIeasbase.tCIp11.4.Fundamentallocustheorems.Thefollowingthreetheoremscan,PA-I///"{easilybediscoveredandprovedbythestudent.Theproofswillbeleftto/f,///1'""thestudent./",/............/,/----........../",I/"",pI//'"//"AM"BTheorem11.1.Thelocusofpoints/~1M1inaplaneatagivendistancefromI/dfDafixedpointisacirclewhosecenterJJDisthegivenpointandwhoseradius/Theorem11.4.measureisthegivendistance.'./""""'/PARTI:Anypointontheperpendicularbisectorofthelinesegmentjoiningtwopointsisequidistantfromthetwopoints.Theorem11.1.Given:Ci51.AB;AM=MB;PisanypointonCD,I'¥=M.'II!--Conclusion:AI'=BP.Theorem11.2.Thelocusofpoints-:r------.IProofIIinaplaneatagivendistancefromadIgivenlineintheplaneisapairof,-:.1-,-t.-STATEMENTSREASONSlinesintheplaneandparalleltothedgivenlineandatthegivendistance_--L:--II.CD1.AB;AM=MB.I.Given.fromthegivenline.T:-I2.LAMPandLBMParerightL:i.2.§1.20.ThcM"'''~-j3.DrawPAandPB.3;Postulate2~4.Pi1=PM.4.Theorem4.1.5.DAMP==f':,.BMP.5.Theorem4.13.6.:.AP=BP.6.§4.28.11Theorem11.3.ThelocusofpointsPARTII:Anypointequidistantfromtwopointsliesontheperpendicularbisectorinaplaneequidistantfromtwoofthelinesegmentjoiningthosetwopoints.givenparallellinesisalineparallel-;----f-~lGiven:I'anypointsuchthatAI'=BP;CM1.AB;AM=BM.tothegivenlinesandmidwayConclusion:PliesoneM.betweenthem.ProOfTheorem11.3.STATEMENTSREASONSTheorem11.41.Plies+--,)0onCMorI'doesnotlieonI.Lawofexcludedmiddle..CM.11.5.Thelocusofpointsinaplanewhichareequidistantfromtwogiv<2.AssumeI'doesnotlieonCM.2.Temporaryassumption.pointsintheplaneistheperpendicularbisectorofthelinesegmentjoi3.DrawPM.3.Postulate2.thetwopoints... GEOMETRICLOCI325324FUNDAMENTALSOFCOLLEGEGEOMETRY4.Given.PARTII:Anypointequidistantfromthesidesofanangleisonthebisectorofthe4.AP=BP.5.Given.angle.5.AiH=BM.6.Theorem4.1.6.PM=PM.Given:BFbisectsLABC;PE1.BA;PD1.BC;PE=PD,iJP7.,6.AJvlP==,6.BiHP.7.S.S.S.Conclusion:PliesonlIF.8.LAMP==LBMP.8.§4.28.9.Theorem3.14.9.JWP1.AB.10.Given.Proof10.CM1.AB.11.Therearetwodistinctlinespass-11.Statements9and10.STATEMENTSREASONSingthroughPandperpendiculartoAB.12.Theorem5.2.1.PE1.BA,PD1.BC.1.Given.12.Thisisimpossible.13.Eitherpornot-p;[not(not-p)]~2.LBEPandLBDParerightL§.2.§1.20.13.:.PmustlieonCM.p.3.PE=PD.3.Given.4.BP=BP.4.Theorem4.1.5.,6.BEP==,6.BDP.5.Theorem5.20.6.La==Lf3.6.§4.28.Theorem11.57.:.BPbisectsLABC.7.§1.19.11.6.Thelocusofpointsintheinteriorofananglewhichareequidistantfromthesidesoftheangleisthebisectoroftheangleminusitsendpoint.11.7.Corollary:Thelocusofpointsequidistantfromtwogivenintersect-PARTI:Anypointonthebisectoroftheangleisequidistantfromthesidesofthe.~'inglinesisthepairofperpendicularlineswhichbisectstheverticalanglesangle.formedbythegivenlines.Given:BFbsectsLABC;pointP#BonBF;PE1.1M;PD1.EG.Theorem11.6. :PE=PD.I~&Thi:locu~-~.--4.Thcon:m5.3.!>.2"5.DE+--++-+IICF.5.Theorem5.5.k--3Yo,,Jt6.EFCDisaD.6.DefinitionofaD.EX.B.Ex.9.~6"7.LDEFisarightL.7.Definitionof-LJines.8.EFCDisaD.8.DefinitionofaD.10.WhatistheareaofthecrosssectionoftheI-beamshowninthefigure?9.DE=CF;AD=Be.9.Theorem6.2.11.ComputetheareaofthecrosssectionoftheaccompanyingH-beam.10.6AED==6BFC.10.Theorem5.20.11.AreaEBCD=areaEBCD.11.Reflexiveproperty.]2.AreaEBCD+areaAED=area12.Additiveproperty.EBCD+areaBFC.1--2.5"13.AreaABCD=areaEBCD+area13.Postulate22.3"AED;areaEFCD=areaEBCD+areaBFe.]4.AreaABCD=areaEFCD.14.Substitutionproperty.15.AB=b;DE=h.15.Given.]6.AreaEFCD]=bh.16.Postulate24.~7.AreaABCD=bh.]7.Substitutionproperty.Ex.10.Ex.n.---------------------------- AREASOFPOLYGONS34746FUNDAMENTALSOFCOLLEGEGEOMETRYExercises2.6.Corollary:Parallelogramswithequalbasesandequalaltitudesarequalinarea.1.Findtheareaofaparallelogramthebaseofwhichis16inchesandthealtitudeofwhichis10inches.2.7.Corollary:Theareasoftwoparallelogramshavingequalbaseshave2.Findtheareaofaparallelogramthebaseofwhichis16.4feetandthehesameratioastheiraltitudes;theareasoftwoparallelogr~mshavingaltitudeofwhichis11.6feet.equalaltitudeshavethesameratioastheirbases.3.Findthealtitudeoftheparallelogramtheareaofwhichis204squareinchesandthebaseofwhichis26inches.Theorem12.24.FindtheareaofDABCDifAB=24yards,AD=18yards,andmLA=30.12.8.Theareaofatriangleisequaltoone-halftheproductofitsbaseand5.Findtheareaof0ABCDifAD=12inches,AB=18inches,mLA=60.it~altitude..CD7Gwen:,6.ABCwIthbaseAB=bandcaltitudeCE=h.I///Conclusion:Areaof,6.ABC=tbh.~,:h/'t"///i:Ak~BbI...\Theorem12.2,~J.'.ABProofREASONSExs.4,5.STATEMENTS1.Given.6.Findtheareaoftherhombus,thediagonalsofwhichare35inchesand1.,6.ABChasbase=b,altitudeCE24inches.=h.2.DrawCDIIABandBDIIAC,meet-2.Postulate18;Theorem5.7.7.Given:GF/IAE;CFIIAG;CGIIEF;GF=14inches;BG=12inches;ABing.at.D.=DE=7inches.Findtheareaof(a)0ACFG;(b)0CEFG;(c)tlnitionofaD..'~!uBDFe;(d)L,ABe;(e)L,GCF;(}),6.CEF.4.,6.ABC,6.DCB.4.§6.6.8.FindtheareaofL,RTKifRK=15,KS=12,ST=18.==5.AreaABDC=areaABC+area5.Postulate22.DCB.K6.AreaABDC=areaABC+area6.Substitutionproperty.IABC.7.AreaABDC=bh.7.Theorem12.1.128.2AreaABC=bh.8.Theorem3.5.9.E-7.M~189.Areaof,6.ABC=tbh.ABCDERsTEx.7.Ex.B.12.9.Corollary:Triangleswithequalbasesandequalaltitudesareequalinarea.9.Given:L,ABCwithCD1..AB,AE1..BC,AB=24inches,CD=15inches,12.10.Corollary:TheareasoftwotriangleshavingequalbaseshavetheBC=21inches.FindAE.sameratioastheiraltitudes;theareasoftwotriangleshavingequalaltitudes10.Howlongisthelegofanisoscelesrighttriangletheareaofwhichishavethesameratioastheirbases.64squarefeet?II.FindtheareaoftrapezoidABCD.12.11.Corollary:Theareaofarhombusisequaltoone-halftheproductof12.FindtheareaoftrapezoidRSTQ.itsdiagonals.-----------l... 348FUNDAMENTALSOFCOLLEGEGEOMETRYAREASOFPOLYGONS349CProofSTATEMENTSREASONS14"c1.DrawdiagonalACdividing1.Postulate2.thetrapezoidinto&ABCand}-ACD.AiB2.Areaoff::::.ABC=tblh.2.Theorem12.2.AB18"3.Areaoff::::.DAC=tb2h.3.Theorem12.2.Ex.9.Ex.n.4.Areaoff::::.ABC+areaoff::::.DAC4.£-4.=tb]h+tb2h=th(bl+b2).5.Areaoff::::.ABC+areaoff::::.DAC5.Postulate22.=areaoftrapezoidABCD.6.:.AreaoftrapezoidABCD=6.Theorem3.5.th(bl+b2).R24's12.13.Otherformulasfortriangles.InthissectionweshalldevelopsomeEx.12.importantformulaswhichareusedquiteextensivelyinsolutionsofgeometricproblems.Itisassumedinthisdiscussionthatthestudenthasabasicknow-ledgeofsquarerootsandradicals.Atableofsquarerootscanbefoundon13.Given:DisthemidpointofAF,areaofDABCD=36squareinches.Ipage421ofthistext.Findtheareaoff::::.ABF.Jj14.KisthemidpointofRS.Areaoff::::.RST=30squareinches.Areaoff::::.RKL=7squareinches.Findareaoff::::.LST..~FormulaI:Relatethediagonaldandthesidesofasquare.FTd2=S2+S2=2S2~s:.d=sV2A~BRKSEx.13.Ex.14.Theorem12.3s12.12.TheareaofatrapezoidisequaltohalftheproductofitsaltitudeandIFormula1.thesumofitsbases.IFormula2:Relatethesidesandthediagonaldofasquare.i~Given:TrapezoidABCDwithaltitudeDb2C'CE=h,baseAB=bl,ands!'2=d(Formula1)baseDC=b2.dConclusion:Areaoftrapezoids=V2ABCD=th(bl+b2).Abl=~xV2=dV2Theorem12.3.v2V22r~ 350FUNDAMENTALSOFCOLLEGEGEOMETRYAREASOFPOLYGONS351D15CD18CFormula3:Relatealtitudehandsidesofanequilateraltriangle.mLRTM=30;TM1-RS.Th2=s2_(~rA~60'B~iAB3S2-Ex.5.Ex.6.4sLLsV3/2h=RM2Formula3.D1'10"CFormula4:RelatetheareaAandthesidesofanequilateraltriangle.20c18/Q,tA=tRSXTMA~JBABlxsV34'3"=ts2.1Ex.2.E~8,--------------------------------------------------------------s2Y3y=4.9-14.Solveforxineachofthefollowingfigures.ExercisesDC1-8.Findtheareaofeachofthefollowingtrapezoids.I;KT~ID16C:8'~~oD15CA12BR24S,Ex.9.Given:ABCDisa0.Ex.10.Given:KRSTisa0.!:5.Air~BA(24~BE26Ex.1.Ex.2.cTN8D10CAi~RA~~BL20ABEX.3.Ex.4.Ex.11.Given:L/HNTisaD.Ex.12.F~~ccc~c----------------------------------------------------------------- AREASOFPOLYGONS353352FUNDAMENTALSOFCOLLEGEGEOMETRYJItcanbeshownthattheratioofthecircumferenceofacircletoitsdiameterisaconstant.ThisconstantisrepresentedbytheGreekletter1T(Pi).Thus,inCFig.12.8,CIID!=C2/D2=1T,12.15.Historicalnoteon'1T.Theabovefactwasknowninantiquity.~lOVariousvalues,astoundinglyaccurate,werefoundbytheancientsforthisconstant.~Perhapsthefirstrecordofanattempttoevaluate1TiscreditedtoanHKIAxBEgyptiannamedAhmes,about1600B.C.Hisevaluationof1Twas3.1605.Ex.13.Ex.14.Archimedes(287-212B.C.)estimatedthevalueof1Tbyinscribinginandcir-cumscribingaboutacircleregularpolygonsof96sides.Hethencalculated15-16.Findtheareasofthefollowingequilateraltriangles.theperimetersoftheregularpolygonsandreasonedthatthecircumferenceCNofthecirclewouldliebetweenthetwocalculatedvalues.Fromtheseresultsheprovedthat1Tliesbetween3+and3j~.Thiswouldplace1Tbetween3.1429and3.1408.Ptolemy(?1O0-168A.D,)evaluated1Tas3.14166.Vieta(1540-1603)gave3.141592653asthevalueof1T.Studentsofcalculuscanprovethat1Tiswhatistermedanirrationalnumber;i.e.,nomattertowhatdegreeofaccuracytheconstantiscarried,itwillnever5beexact.Itcanbeshowninadvancedmathematicsworkthat1T=4(1ADB-i+!LM-t+t-A...).Theright-handexpressioniswhatistermedaninfiniteEx.15.Ex.16.senes.Byusingthemoderncalculatingmachinesoftoday,thevalueof1Thasbeenfoundaccuratetomorethan100,000digits.Thisisadegreeofaccuracy17.Theareaofatrapezoidis76squareinches.Thebasesare11inchesand8inches.Findthealtitude.1whichhasnopracticalvalue.Thevalueof1Taccurateto10decimalplacesis,',~),1415926536.18.Findtheareaoftheisoscelestrapezoidthebasesofwhichare9yards-.--,-'.-and23yard:nttldthediagonalofwhichi£11~¥ards'nun12.16.Ciiciiriiferenceofa-C1rae:~Sriice-T71r~-1T,-we-c-an--riowderivca19.Findthediagonalofasquareofside(a)6inches;(b)15inches.--'formulaforthecircumference.Ifwemultiplyeachsideoftheequationby20,Findthesideofasquarethediagonalofwhichis(a)16inches;(b)32,:D,weobtainC=1TD.SincethediameterDequalstwotimestheradius,weinches,cansubstituteintheequationandgetC=21TR.21.Findtheareaofasquarethediagonalofwhichis(a)12inches;(b)52Thus,thecircumferenceofacircleisexpressedbytheformulaC=1TD,orC=21TR.inches.22.Findthealtitudeofanequilateraltrianglethesideofwhichis(a)8inches;i~,3(b)17inches,23.Findtheareaofanequilateraltrianglethesideofwhichis(a)7inches;(b)52inches.12.14.Formulasforthecircle.Findingthelengthandareaofacirclehavebeentwoofthegreathistoricproblemsinmathematics.Inthistextwewill~notattempttoprovetheformulasforthecircle.Thestudenthasalready:,courses.Jhadmanyoccasionstousetheseformulasinhisothermathematics0Wewillreviewtheseformulasandusetheminsolvingproblems.;~~Definition:Thecircumferenceofacircleisthelengthofthecircle(some-'1Fig.12.8.timescalleditsperif!leter).- AREASOFPOLYGONS355354FUNDAMENTALSOFCOLLEGEGEOMETRY3.Findthediameterofacirclethecircumferenceofwhichis(a)280;Inusingthisformula,itisbesttouse1Troundedofftoonemoredigitthan(b)87.54;(c)68.3562.thedata,DorR,itisusedwith.Thenroundofftheanswertothedegreeof4.Findtheradiusofacirclethecircumferenceofwhichis(a)140;(b)26.38;accuracyofthegivendata.(c)86.6512.12.17.IllustrativeExample1.Findthecircumferenceofacirclethedia-5.Findtheradiusofacircletheareaofwhichis(a)24.5;(b)37.843;(c)meterofwhichis5.7inches.913.254.Solution:C=1TD6.Findthediameterofacircletheareaofwhichis(a)376;(b)62.348;C=3.14(5.7)(c)101.307.=17.8987.Iftheradiioftwocirclesare2and3inchesrespectively,whatistheratioAnswer:17.9inches.oftheirareas?12.18.IllustrativeExample2.Findtheradiusofacirclethecircum-8.Iftheradiioftwocirclesare3and5inchesrespectively,whatistheratioferenceofwhichis8.25feet.oftheirareas?9.Findtheareaofacirclethecircumferenceofwhichis28.7feet.Solution:C=21TR10.FindtheperimeterofthetrackABCDEFinthefigure.R=~11.FindtheareaenclosedbythetrackABCDEFinthefigure.21TE440'D8.25-2(3.142)=1.313FcAnswer:1.31feet.'12.19.Areaofacircle.ItcanbeshownthattheratiooftheareaofacircleABtothesquareofitsradiusisaconstant.Thisisthesameconstant1T,whichExs.10,11.equalstheratioofthecircumferenceandthediameterofacircle.InFi~.12,9.12.FindtheareaofasernicirclethepCl:imctcl'ofwhichi£36.43.;113.Fmdtheshadedarea.14.Findtheareaoftheshadedportionofthesemicircle.",i!LA2.--=1TR2/:.,RJ2R22,.1,,8,.,.,:,,~':1~Fig.12,9.0Thus,theareaofacircleisgivenbytheformulaA=1TR2.SinceR=D/2,wecansubstituteintheformulaandgetA=(1TD2/4).Ex.13.Ex.14.Exercises15.Findtheareaoftheshadedportioninthefigure.1.Findthecircumferenceofacircletheradiusofwhichis(a)5.2;(b)2.54;;.~16.Findtheperimeterofthefigure.(c)32.58.17.Findthetotalareaenclosedinthefigure.2.Findtheareaofacircletheradiusofwhichis(a)7.0;(b)8.34;(c)25.63.r... SummaryTestsEx.15.Ex.16,1718.Findthelengthofthebeltusedjoiningwheels0and0'.Test1COMPLETIONSTATEMENTS1.Theratioofthecircumferencetothediameterofacircleis2.Theareaofarhombusisequaltoone-halftheproductofoftherhombus.3.Theareaofanequilateraltrianglewithsidesequalto8inchesissquareinches.I4.Theareaofthesquarewiththediagonaleqmilt~_~il~chesis,,-klQ"i,n.nnhe---15"--;_1squareinches.',..5.Thenumber1Trepresentstheratiooftheareaofacircletoits-Ex.lB.6.Ifthealtitudeofatriangleisdoubledwhilethearearemainsconstant,thebaseismultipliedby19.Findtheareaoftheshadedportion.20.Findthetotalareaenclosedinthefigure,giventhatthe,idc,ofthe'7.Doublingthediameterofacirclewillmultiplythecircumferencebyequilateral6arediametersofthesemicircles.8.Theratioofthecircumferenceofacircletotheperimeterofitsinscribedsquareis24"Test21s.rATRUE-FALSESTATEMENTS1.Themedianofatriangledividesitintotwotriangleswithequalareas.2.Tworectangleswithequalareashaveequalperimeters.36"3.Theareaofacircleisequalto21TR2.4.Iftheradiusofacircleisdoubled,itsareaisdoubled.Ex.19.Ex.20.356357------------------------------------Ji8-----~~ SUMMARYTESTS359358FUNDAMENTALSOFCOLLEGEGEOMETRY5.Theareaofatrapezoidisequaltotheproductofitsaltitudeandits20"median.6.Triangleswithequalaltitudesandequalbaseshavethesameareas.16,/14.Findtheareaofthetrapezoid.7.Theareaofatriangleisequaltohalftheproductofthebaseandoneofthesides.~8.Ifthebaseofarectangleisdoubledwhilethealtituderemainsunchanged,theareaisdoubled.Prob.4.9.Asquarewithaperimeterequaltothecircumferenceofacirclehasanareaequaltothatofthecircle.5.Findtheshadedarea.10.Doublingtheradiusofacirclewilldoubleitscircumference.6.Findtheshadedarea.11.Ifatriangleandaparallelogramhavethesamebaseandthesamearea,theiraltitudesarethesame.12.Thelinejoiningthemidpointsoftwoadjacentsidesofaparallelogram12'cutsoffatriangletheareaofwhichisequaltoone-eighththatoftheparallelogram.13.Thesumofthelengthsoftwoperpendicularsdrawnfromanypointonthebasetothelegsofanisoscelestriangleisequaltothealtitudeonaleg."'.i""...-Test320'Prob.5.Prob.6.PROBLEMSIi1.FindtheareaoftheI-beam.7.Findtheperimeterofthefigure.2.Findtheareaofthetrapezoid.I8.Findtheratioofthecircumferencesof00andQ.--~;.''r-I~~"--'.'l.':'1~;:_,,',,:,----...-----8"I"....0--~"Dt'2"324'l40'ItProb.2.~Prob.1.~30'Prob.7.Prob.8.3.Findtheareaofthetriangle.Prob.3.12'----- ahalf-linewhichdoesnotincludethenumber-Iasamemberofthesetit1131represents.InFig.13.3,weseethegraphofthesetofallrealnumbersfrom-1to2inclusive.MathematicianshavedevelopedaconcisewaytodescribesetssuchastheoneshowninFig.13.3.Itiswritten{xl-I:s::x:s::2}andisread"thesetofallrealnumbersxsuchthatx~-1andx:s::2."...:II.AII,BI)-3-2-10123Fig.13.3.{xl-I'"x2}.'"CoordinateGeometryTheintersectionofthesetofallrealnumbersxsuchthatxislessthan2andthesetofallrealnumbersxsuchthatxisgreaterthanorequalto-1canbewrittenas{xix<2}n{xix~-I}.ThegraphofsuchasetisshowninFig.13.4.~o13.1.Thenatureofcoordinategeometry.Uptothispointthestudenthas0-+receivedtraininginalgebraandgeometry,butprobablyhadlittleoccasionto1-"""""-LLA,LBLstudytherelationshipbetweenthetwo.----.0.-3-2-1023Intheyear1637theFrenchphilosopherandmathematicianReneDescartes''..'(1596-1650)establishedalandmarkinthefieldofmathematicswhenhis,Fig.13.4.{xix<2}n{xix~I}.,1,bookLaGeometriewaspublished.Inthisbookheshowedtheconnection'.-betweenalgebraandgeometry.Byestablishingthisrelationship,hewasable'Exercises'.'tostudygeometricfiguresbyexaminingvariousequationswhichrepresented,:..,,.,..-InExercjs~sI-:-t.>...usesetn()ta~i-I}.Noticethattheopendotindicatesthegraphtobe'Ex.3.4.IIbIIII)L.L1.L;>0"-3-2-10123-3-2-1023Fig.13.2.{xix>-I}.Ex.4.360361~l.. COORDINATEGEOMETRY36362FUNDAMENTALSOFCOLLEGEGEOMETRYyL~\II..023-3-2-14Ex.5.1L~3L..6.~-2-1023MEx.6.2f-=--------1P{3,2)IIInExercises7-12,drawagraphforeachset.If-VI8.{xix~2}.I7.{xx<-2}.10.{xl-4I}14.A={xix>2}B={xix~4}.-2B={xix>-2}.15.A={xixl}.16.A={xix<-l}-3B={xix<-I}.17.A={xlx-I}.20.A={xixYI)andK=(X2,YI),findthecoordinatesofthemidpointofHK.20.1f5=(XbYl)andT=(Xl')2),findthecoordinatesofthemidpointof5T.13.5.Thedistanceformula.S~osePandQaretwopointswithco-ordinates(xp,yp)and(xQ,YQ)'IfPQisnotperpendiculartoeitherthex-axis,Exs.2-5.orthey-axis,thenperpendicularsfromPandQtothecoordinateaxeswillintersectatRand5(Fig.13.7).ItcanbereadilyprovedthatL:.PRQisaright4.UsingthefigureofEx.2,butlettingau=5,completethetable.triangle.(Why?)PQisthehypotenuseofrightL:.PRQ..,~1'~~iy.';;'ABCDEPIP2P3P4I"',if,"'s.'~~{H---lQ(xQ'YQ)x-coordinate:~II.v-coordinateIIIIIII5.UsingthefigureforEx.2,ifau=1,find(a)aA;(b)UC;(c)BU;(d)aD;Kf---jR(e)AB;(j)CD;(g)CPI;(h)EPI;(i)HP2;(j)BP2;(k)FPa;(l)GP4.t(xp,yp)I6.IfA=(0,0),B=(3,0),C=(4,5),D=(1,2),plotthesetofallpointswhichbelongtopolygonABCD.0x7.PlotthepointsR(1,2),5(-1,1),T(-I,3).IndicatethepointswhichlieintheinteriorofL:.R5T.8.Describethesetofallpointsforwhichthex-coordinateis2.9.Describethesetofallpointsforwhichthey-coordinateis-3.10.Plotthesetofallpoints(x,Y)forwhichxandyareintegerswhichsatisfythefollowingconditions.Fig.13.7.[. 368COORDINATEGEOMETRY369FUNDAMENTALSOFCOLLEGEGEOMETRYB(XB)YBBythePythagoreantheorem,'(PQ)2=(PR)2+(RQ)2Q(XB'YM)orPQ=Y(PR)2+(RQ)2A(xA'YA)P(XM'YA)But,PR=MN=XQ-XpandRQ=KL=YQ-YP'Usingthesubstitutionxpropertyofequality,wegetAl0BIPQ=Y(xQ-xp)2+(YQ-YpF(Yp-YQ)2,wecanstatetheSince(XQ-Xp)2=(Xp-XQ)2and(YQ-Yp)2=following..'~'Fig.13.8,Theorem13.1(Thedistanceformula).ForanytwopointsPandQ.:"~''.'-1'"~f.::..Similarly,bysettingPM=QB,weget:,,.,V(Xp-XQ)2(Yp-YQ)2.'"PQ=V(XQ-Xp)2+(YQ-Yp)2=.,._-I1.YA+YBThestudentshouldconvincehimselfthatthedistanceformulawillalso'"'IYM--'2;,''"workifthesegmentisperpendiculartoeitheraxis...1,,'Example:FindthedistancebetweenPt(-3,-4)andP2(2,-7).,Theorem13.2(themidpointformula).MisthemidpointofABifandonlySolution:UsingTheorem13.1,weget"ifXu=t(XA+XB)andYM=t(YA+YB)'.'.'.'ExamPle.FindthelengthofthemedianfromvertexBofL.ABCwiththePtP2=Y[2-(-3)]2+[-7-(-4)]2.'.,..followingvertices:A(-I,I),B(3,4),C(5,-7).,1::,...-,=/~.;:.!=Y3413.6.Midpointofasegment.Quitefrequentlywewillwishtofindthe'j',,coof,~>,.Y.Y.,18.Thediagonalsofasquareintersectatrightangles.1",I,",,,,"',"''.Yy.'",',~.,,,,D(d,c)C(b,c)D(0,b)C(b,b)1,,,0xx0A(0,0)B(b,0)xxA(O,0)B(a,0)',uiExample(a).Example(b).-Ex.n.Ex.18.Y'~~Ii:'19.Thelinesjoiningthemidpointsoftheoppositesidesofaquadrilateral,bisecteachother.20.Thediagonalsofarhombusintersectatrightangles..y.Yx0C(a+b,c)H(a,bJxxA(0,0)Ex.20.ExamPle(c).Ex.19.-------------------------.. .-.378FUNDAMENTALSOFCOLLEGEGEOMETRY,iCOORDINATEGEOMETRY379Y~Exercises'¥cDrawanddescribethegraphswhichsatisfythefollowingconditions.1.x~O.2.y0andy>O.4.x<0andy>O.-1ax5.x>0andyOorx<-2.8.Y<1ory>4.9.-12.15.x>2andy<-1.16.x>O,y>O,andy=x.y.....13.11.Equationofaline.Theequationofalineinaplaneisanequation':intwovariables,suchasxandy,whichissatisfiedbyeverypointontheline?~:'..1'.andisnotsatisfiedbyanypointnotontheline.Theformoftheequation~",,.,:willdependuponthedatausedindeterminingtheline.Astraightlineis.I.::,.determinedgeometricallyinseveralways.Iftwopointsareusedtodeter-.minetheline,theequationofthelinewillhaveadifferentformthanifonexpointandadirectionwereused.Wewillconsidersomeofthemorecommonaformsoftheequationforastraightline.13.12.Horizontalandverticallines.IfalineIIisparalleltothey-axis,theneverypointonIIhasthesamex-coordinate(seeFig.13.13).Ifthisx-coordinateisa,thenthepointP(x,y)isonIIifandonlyifx=a.Fxrrmple(e).Inlikemanner.y=bistheequationof1"2'alinethrough(0,b)paralleltothex-axis.yxFig.13.13.13.13.Point-slopeformofequationofaline.OneofthesimplestwaysinwhichalineisdeterminedistoknowthecoordinatesofapointthroughExample({).whichitpassesandtheslopeoftheline.~-----------Ii 380FUNDAMENTALSOFCOLLEGEGEOMETRYCOORDINATEGEOMETRY381ConsideranonverticallineIpassingthroughPI(XbYI)withaslopemSolution:Wemustfirstreduceourequationtothepoint-slopeform,as(seeFig.13.14).LetP(x,y)beanypointotherthanPlanthegivenline.p'follows:y5x-2y=115x-11=2y(Why?)2y=5x-II(Why?)=5(x-V)y=t(x-V)(Why?)xory-O=t(x-li)Comparingthisequationwiththestandardpoint-slopeformequation,wefindthatthelinepassesthrough(¥,0)andhasaslopeon.13.14.Two-pointformofequationofaline.Theequationofastraightlinethatpassesthroughtwopointscanbeobtainedbyuseofthepoint-slopeformFig.13.14.andtheequationfortheslopeofalinethroughtwopoints.Thus,ifPI(Xb1"1)andP2(X2'Y2)arecoordinatesoftwopointsthroughwhichthelinewilllieonIifandonlyiftheslopeofWIism;thatisP(x,y)isonI~slopeofpasses,theslopeofthelineism=(YI-Y2)/(XI-X2)andsubstitutingthisWIism,orvalueforminthepoint-slopeform,wegettheequationYI-Y2y-yY-YI=-(X-XI)P(x,y)isonl~~=mXI-X2X-XIThisiscalledthetu'o-pointformoftheequationofastraightline.and.~~?ExercisesP(X,y)isonI~Y-YI=m(x-xI)'Findanequationforeachofthelinesdescribed.Theorem13.6.ForeachpointPI(XbYI)andforeachnumberm,theequation1.Thelinecontainsthepointwithcoordinates(7,3)anditsslopeis4.ofthelinethroughPIwithslopemisY-YI=m(x-XI)'Thisequationiscalledthe2.Thelinecontainsthepointwithcoordinates(-2,-5)andhasaslopeof3.point-slopeformofanequationofaline.3.Thelinehasaslopeof-2andpassesthrough(-6,8).4.Thelinehasaninclinationof45andpassesthrough(3,5).ExamPle.Findtheequationofthelinewhichcontainsthepointwithco-5.Thelinepassesthrough(-9,-3)andisparalleltothex-axis.ordinates(2,-3)andhasaslopeof5.6.Thelinecontainsthepoint(5,-7)andisperpendiculartothex-axis.Solution:7.Thelinecontainsthepointswithcoordinates(4,7)and(6,11).8.Thelinecontainsthepointwhosecoordinatesare(-1,1)andhasanY-YI=m(x-xI)inclinationof90.Y-(-3)=5(x-2)y+3=5x-10InExs.9-16findtheslopeofeachofthelineswiththefollowingequations.5x-y-I3=09.3x-y=7.10.2x+y=8.Example.Findtheslopeofthelinewhoseequationis5x-2y=11.11.5x+3y=9.12.Y=x.I,.. 382FUNDAMENTALSOFCOLLEGEGEOMETRYCOORDINATEGEOMETRY38313.2x=y.14.y=2x-7.b-Oy-b=-(x-O)15.y=-3.16.x=6.orO-aInExs.17-22,pointsA(-2,4),B(2,-4),C(6,6)areverticesof,6,ABe.y-b=_b-x17.Findtheequationofn.a18.FindtheequationofBC.Thismaybereducedto19.FindtheequationofthemediandrawnfromC.'1120.FindtheequationofthemediandrawnfromA.':ibx+ay=ab21.FindtheequationoftheperpendicularbisectorofAB.andbydividingbothsidesoftheequationbyab,weget22.FindtheequationoftheperpendicularbisectorofBe.,..;;~xYyC(6,6)~,.a-+-b=1~,:r(I".'';whichistheinterceptformoftheequationofaline.'.''.'c'~.,,":':,II'...,.'ExamPle.DrawthegraphofthelineLwhoseequationis3x-4y-12=O.:,,.~I.Solution:3x-4y-12=O.xX'Adding12tobothsides,3x-4y=12.Dividingbothsidesby12,::_~]=4~Y'orxYExs.17-22.4+(-3)=113.15.Interceptformofequationofaline.Thex-interceptandy-interceptofalinearedefinedasthecoordinatesyHencethex-interceptis4andthey-interceptis-3.ofthepointswherethelinecrossesthe13.16.Slopeandy-interceptformoftheequationofaline.Ifthey-inter-x-axisandthey-axisrespectively.Theceptofalineisbandtheslopeofthelineism,wecandeterminetheequationtermsarealsousedforthedistancesofthelinebyusingthepoint-slopeform.Thus,thesepointsarefromtheorigin.Thecontextofthestatementwillmakeclear0xy-b=m(x-O)ifacoordinateordistanceismeant.Ifthex-interceptandy-interceptofaorlinearerespectivelyaandb(Fig.13.15),thecoordinatesofthepointsofinter-y=mx+b"";":sectionofthelineandaxesare(a,O)1i'"and(0,b).Usingthetwo-pointform,Fig.13.15.iThisiscalledtheslopey-interceptform.'I,:.1",.,..".":"°"-----------.I....6, 384FUNDAMENTALSOFCOLLEGEGEOMETRYCOORDINATEGEOMETRY385ExercisesTheorem13.7.ThegraPhoftheequationy=mx+bisthelinewithslopemandy-interceptb.Whatarethex-andy-interceptsofthegraphsofthefollowingequations?Whataretheslopesofeachequation?Drawgraphsoftheequations.Example.Whatistheslopeofthelinewhoseequationis2x-5y-17=O.Solution:Reducetheequationtoy=mx+bform,asfollows1.3x+4y=12.2.2x-3y-6=O.2x-5y-17=03.x+5=O.2x-I7=5y(Why?)4.y-7=o.5y=2x-I7(Why?)Determinetheequations,inAx+By+C=0form,ofwhichthefollowingy=~x-~(Why?)linesaregraphs.Theslopeofthelineis~.5.Thelinethrough(2,3)withslope5.6.Thelinethrough(-5,1)withslope7.13.17.Thegeneralformoftheequationofaline.Themostgeneralform7.Thelinethrough(4,-3)withslope-2.ofanequationofthefirstdegreeinxandyis8.Thelinethrough(1,1)and(4,6).Ax+By+C=09.Thelinethrough(2,-3)and(0,-9).10.Thelinethrough(-10,-7)and(-6,-2).II.Thelinewithy-intercept5andslope2.whereAandBarenotbothzero.ii,!12.Thelinewithy-intercept-3andslope1.!IfBop0,wecansolveforytoget13.Thelinewithy-intercept-4andslope-3..''14.Thex-axis.AC,1.',:,y=--x--'IS.They-axis.-BB'16.Theverticallinethrough(-5,7).17.Theverticallinethrough(3,-8).Thi,equationh"thefonny~mx+band,hence,mu"hetheeqoarionof.1JR.Thelinethrough(2,5)andparalleltothelinepassingthrough(-2,-4)straightlinewith""PhiZ~Zeta-=/;XiXXChiHYJEta00Omicron'¥t/JPsief:IThetaII7TPin(J)OmegaSYMBOLSANDABBREVIATIONS~.HcaningAUBtheunionofsetsAandBAnBtheintersectionofsetsAandBA'thecomplementofsetAACBAisasubsetofBxEBxisamemberofsetBx~BxisnotamemberofsetBjJlqpandqPVqporq(inclusive)P~qporq(exclusive)jJ~qpimpliesq;ifp,thenqjJ~qpisequivalenttoq;pif,andonlyif,qiffif,andonlyif{}notationforset(}nullsetAl1intervalABABsegmentABI419r.. 420FUNDAMENTALSOFCOLLEGEGEOMETRYTABLEI.SQUAREROOTSABlineAB.'jNyNNyNNyNNyNNyNNyNlIBhalf-lineAB11.000517.14110110.05015112.28820114.17725115.843ABrayAB21.414527.21110210.10015212.32920214.21325215.875I,1;31.732537.28010310.14915312.36920314.24825315.906mABthemeasureofsegmentABf£42.000547.34810410.19815412.41020414.28325415.937ABthemeasureofsegmentABp';~52.236557.41610510.24715512.45020514.31825515.969,iLangle,662.449567.48310610.29615612.49020614.35325616.000i§angles.'72.646577.55010710.34415712.53020714.38725716.03182.828587.61610810.39215812.57020814.42225816.062mLABCthemeasureofangleABC'~--93.000597.68110910.44015912.61020914.45725916.093ABarcAB103.162607.74611010.48816012.64921014.49126016.125mABthedegreemeasureofarcAB113.317617.81011110.53616112.68921114.52626116.155-isequalto;equals123.464627.87411210.58316212.72821214.56026216.186¥-isnotequalto133.606637.93711310.63016312.76721314.59526316.217y;thenonnegativesquarerootofx143.742648.00011410.67716412.80621414.62926416.248153.873658.06211510.72416512.84521514.66326516.279IxItheabsolutevalueofx.164.000668.12411610.77016612.88421614.69726616.310---..-issimilarto174.123678.18511710.81716712.92321714.73126716.340184.243688.24611810.86316812.96121814.76526816.3711-isperpendicularto194.359698.30711910.90916913.00021914.79926916.401isparalleltoII204.472708.36712010.95417013.03822014.83227016.432>isgreaterthan214.583718.42612111.00017113.07722114.86627116.4620)~e>a(p.73),PropertiesofaFieldF-I(closurepropertyforaddition):a+bisauniquerealnumber(p.73).F-2(associativepropertyforaddition):(a+b)+e=a+(b+c)(p.73).F-3(commutativepropertyforaddition):a+b=b+a(p.73).PROPERTIESOFREALNUMBERSYSTEMF-4(additivepropertyofzero):Thereisauniquenumber0,theadditiveidentityelement,suchthata+0=0+a=a(p.73).F-5(additiveinverseproperty):Foreveryrealnumberathereexistsarealnumber-a,theadditiveinverseofa,suchthata+(-a)(p.73).=(-a)+a=0EqualityPropertiesF-6(closurepropertyformultiplication):a.bisauniquerealnumber(p.7,)).E-I(reflexiveproperty):a=a(p.72).F-7(associativepropertyformultiplication):(a.b).E-2(symmetricproperty):a=b~b=a(p.72).e=a.(b.c)(p.73).F-8(commutativepropertyformultiplication):a.b=b.a(p.74).E-3(transitiveproperty):(a=b)1(b=c)~a=e(p.72).IF-9(multiplicativepropertyofI):ThereisauniquerealnumberI,theE-4(additionproperty):(a=b)1(e=d)~(a+e)=(b+d)(p.72).multiplicativeidentityelement,suchthata.,I=I.a=a(p.74).E-5(subtractionproperty):(a=b)1(e=d)~(a-c)=(b-d)(p.72).F-IO(multiplicativeinverseproperty):Foreveryrealnumbera(a¥=0),E-6(multiplicationproperty):(a=b)1(e=d)~ae=bd(p.72).fI'thereisauniquerealnumberI/a,themultiplicativeinverseofa,suchthata.(ljt4-=--(ltaJ-n'-4=-1--{p.74).m.-E-7(divisionproperty):«(1=b)1(e=d¥=0)-4=(p.72).~~F-ll(distributiveproperty):a(b+e)=a'b+a'e(p.74).E-8(substitutionproperty):Anyexpressionmaybereplacedbyanequivalentexpressioninanequationwithoutdestroyingthetruthvalueoftheequation(p.73).LISTOFPOSTULATESOrderProperties1.Alinecontainsatleasttwopoints;aplanecontainsatleastthreepointsnotallcollinear;andspacecontainsatleastfourpointsnotallcoplanar0-1(trichotomyproperty):Foreverypairofrealnumbers,aandb,exactly(p.76).oneofthefollowingistrue:ab(p.73).2.Foreverytwodistinctpoints,thereisexactlyonelinethatcontainsboth0-2(additionproperty):(a(e-b)(p.73).,thatcontainsthethreepoints(p.76).0-4(multiplicationproperty):(a0)~aebe(p.73).arepointsoftheplane(p.76).0-5(divisionproperty):(a0)~aleelb(p.73).5.Iftwodistinctplanesintersect,theirintersectionisoneandonlyoneline(able1cia23.LABEandLEBD;LACEandLDCE33.=35.<25.(1)150(b)135(c)90180-x37.1039.227.LABD==LCBD29.AE==CE;BE==DETest231.CD==BDPages44-46I.T3.T5.T7.T9.F11.F13.F15.T17.F19.F21.T23.F25.F27.T29.F31.F33.F35.F37.F39.T41.T43.T45.T47.TExercises(A)49.FI.noconclusion;dogmaybebarkingforareasonotherthanthepresenceofaTest3stranger.3.MarySmithmusttakeanorientationclass.5.noconclusion;thegivenstatementdocsnotindicatethatonlycollegestudentswillI.33.45..~7.I9.50II.2013.55beadmittedfree..15.llO17.95;9519.557.Mr.SmithisacitizenoftheUnitedStates.9.BillSmithwillnotpassgeometry.Pages52-5311.noconclusion;thegivenstatementdoesnotindicatethatonlythosewhoeatZeppocerealarealertonthediamond.13.Itisnotcustomarytoburylivingpersons.Exercises(A)15.Afive-centandfifty-centpiece.17.Itisnotstatedthatthemenplayedfivegamesagainsteachother.I.no3.yes5.yes7.yes9.yes11.yes15.yes19.Two.21.Coinsarenotstampedinadvanceofanuncertaindate.Exercises(B)Exercises(B)1.Itishot.Iamtired.I.BobisheavierthanJack.3.Hisactionwasdeliberate.Hisactionwascareless.3.';[isfortunewillbefallMr.Grimes.5.Thefigureisnotasquare.Thefigureisnotarectangle.5.Iwillgetawartonmyhand.7.Heisclever.Iamnotclever.7.(a)yes;(b)yes;(c)doesn'tlogicallyfollow;(d)nottrue9.SuedislikesKay.KaydislikesSue.9.(a)yes;(b)yes;(c)doesn'tlogicallyfollow;(d)doesn'tlogicallyfollow.II.Twolinesintersect.Twolinesareparallel.':''~..,'~.'..-l~ ,444FUNDAMENTALSOFCOLLEGEGEOMETRYANSWERSTOEXERCISES4457.Thetwolinesarenotparallel;thetwolinesintersect.13.Theanimalisamale.Theanimalisafemale.9.Thenumbersarenaturalnumbers;thenumbersareeitherevenorodd.15.Iwouldbuythecar.Thecarcoststoomuch.11.Itisaparallelogram;itisaquadrilateral.Page5513.Itisabird;itdoesnothavefourfeet.15.Hestudies;hewillpassthiscourse.17.Thepersonsteals;thepersonwillbecaught.I.true;true3.false;true5.false;false7.false;true9.false;false19.Heisaworker;hewillbeasuccess.11.false;true13.false;true15.true;true21.Ihaveyourlooks;Iwillbeamoviestar.Pages56-57Page631.Goldisheavy.1.BobisheavierthanJack.3.Noteveryonewhowantsagoodgradeinthiscourseneedstostudyhard.3.Mydogdoesnotbite.5.Ahexagondoesnothavesevensides.5.FigureABCDisaquadrilateral.7.Everybankerisrich.7.a+c=b+c.9.Twoplus4doesnotequal8.9.Noconclusion.11.Notallequilateraltrianglesareequiangular.11.Iwillgetwartsonmyhand.13.Noblindmencarrywhitecanes.13..JoneslivesinHouston.15.Notallthesecookiesaredelicious.15.a¥b.17.NoteveryEuropeanlivesinEurope.17.Noconclusion.19.Therearenogirlsintheclass.19.y=4.21.Everyquestioncanbeanswered.21.a¥b.23.NoteveryZEPisaz~p.23.Siliff.25.Itisnottmethatanullsetisasubsetofitself.25.Iflisnotparalleltom,thenlnm¥0.Pages58-59Pages64-651.Anapricotisnotafruitoracarrotisnotavegetable.False.3.Nomenliketohuntornomenliketofish.False.1.True.Vegetablesarecarrots.False.5.Somenumbersareoddornoteverynumberiseven.True.3.True.CarsareFords.Flase.7.Thesidesofarightanglearenotperpendicularornotallrightanglesarecon-5.False.Ifheisnotapoorspeller,thenheisajournalist.False.gruent.False.7.Don'tknow.Ifheisamoron,thenhewillacceptyouroffer.Dont'know.9.Noteverytrianglehasarightangleornoteverytrianglehasanacuteangle.9.Don'tknow.Ifapersonstudies,thenhewillsucceedinschool.False.True.11.True.Ifitishard,thenitisadiamond.False.11.Noteverytrianglehasarightangleandnoteverytrianglehasanobtuseangle.13.True.Ifithasthreecongruentsides,thenitisanequilateraltriangle.True,True.ifyouaretalkingabouttriangles;otherwiseitisfalse.13.Notriangleshavethreeacuteanglesornonehaveonlytwoacuteangles.False.15.True.Ifxislargerthany,thenx-y=1.False.15.Araydoesnothaveoneendpointandasegmentdoesnothavetwoendpoints.17.True(?).IfhelivesinCalifornia,thenhelivesinLosAngeles.False.False.19.True.IfX2=25,thenx=5.False.Page60Pages66-671.Premise:Itissnowing.Conclusion:Therainwillbelate.3.Heisacitizen;hehastherighttovote.1.yes3.yes5.yes7.no9.yes11.yes13.no15.yes5.Heisastudent;hemusttakeaphysicalexamination.17.no19.yes21.no(inspacegeometry)23.yes(..Ai 446FUNDAMENTALSOFCOLLEGEGEOMETRYANSWERSTOEXERCISES44';7Pages68-6923.multiplicativepropertyoforder25.given;additionpropertyofequality;subtractionpropertyofequality;division1.(a)1fTE~,thenTERX.propertyofequality(b)IfT~jfX,thenT~RX.27.given;distributiveproperty;additionpropertyofequality;subtractionproperty(c)IfT~RX,thenT~RX.ofequality;divisionpropertyofequality3.(a)IfCEAB,thenCEAlJ.29.given;additionpropertyoforder;subtractionpropertyoforder;divisionpro-(b)IfC~AB,thenC~AB.pertyoforder(c)IfC~An,thenC~AB.5.(a)Ifa=-b,thena+b=O.Pages78-79(b)Ifa~-b,thena+b~O.(c)Ifa+b~0,thena~-b.1.(a)anynaturalnumber(b)one7.(a)If!passthiscourse,thenIhavestudied.3.notnecessarily(b)IfIdonotpassthiscourse,thenIhavenotstudied.5.one(c)IfIdonotstudy,thenIwillnotpassthiscourse.7.four9.(a)Iflinesdonotmeet,thentheyareparallel.9.LineABliesentirelyinoneplane.(b)Iflinesmeet,thentheyarenotparallel.II.yes(c)Iflinesarenotparallel,thentheywillmeet.13.anynonnegativewholenumber,II.(a)Ifthisisnotasquare,thenitisnotarectangle.IS.six(b)Ifthisisasquare,thenitisarectangle.17.yes(c)Ifthisisarectangle,thenitisasquare.19.collinear:d13.(a)Ifthetriangleisequiangular,itisequilateral.coplanarbutnotcollinear:a,b,C(b)Ifthetriangleisnotequiangular,thenitisnotequilateral.notcoplanar:e(c)Ifthetriangleisnotequilateral,thenitisnotequiangular.IS.notvalid17.notvalidPage8119.valid21.validI.B3.55.I23.valid25.notvalid7.89.noII.C13.CIS.AFC17.AED19.7821.42Pages7U-71Pages98-1001.T3.F5.T7.T9.FII.T13.FIS.FTest117.T19.F21.T23.T25.F27.F29.F31.F33.T35.T37.F39.F41.FI.T3.T5.F7.F9.TII.T13.TIS.T17.F19.F21.T23.T(exceptifoneoftheangleshasameasureofzero)25.TPages74-75Test2I.commutativepropertyunderaddition3.additivepropertyofzeroI.postulate3.perpendicular5.obtuse5.distributiveproperty7.1329.congruentII.right7.additionpropertyofequality13.60IS.plane17.complementary9.symmetricpropertyofequality19.lineII.subtractionpropertyofequality13.multiplicationpropertyofequalityPages105-107IS.subtractionpropertyoforderExercises(A)I17.transitivepropertyoforder19.divisionpropertyoforderl.F3.F5.T7.F9.FII.T13.TIS.T21.associativepropertyofmultiplication17.T19.F21.Tiiiiiiiiiiiir---. I,448FUNDAMENTALSOFCOLLEGEGEOMETRYANSWERSTOEXERCISES449Exercises(B)Page14611.qandwaretrue13.(a)(3;t/1(b)noI.AC'=FD.Segmentadditionproperty.3.AG'=GE'=BG'=FG.DefinitionofsegmentbisectorandTheorem4.11.5.LDAG'=LCBE.Anglesubtractionproperty.Pages151-1537.AE'=BG.Transitivepropertyofcongruence.1.'1'3.T5.F7.'I'9.'I'11.F13.F15.FPages109-11017.F19.T21.T23.F1.CHJ-KLM3.ABC-ABCPages168-170CHJ-KA1LABC-ACBCH.J-LKMABC-BAC1.903.1505.457.80CH.J-A1LKABC-BCAGHi-MKLABC-CABPages178-181ABC-CBA5.yes;noTest17.LA-LB9.LFAC-LEBDLADC-LBDCLACF-LBDE1.1803.parallel5.indirect7.perpendicularLACD-LBCDLF-LF9.isosceles11.complementary13.right15.parallelAD-BDAF-BE17.parallel19.obtuse21.vertical;congruentAC-BCAC-BDCD-CDFC-EDTest211.LDAB-LCBA1.F3.F5.F7.F9.'I'11.FLADB-LBCA13.'I'15.'I'LABD-LBAC17.F19.F21.T23.F25.F27.'I'AD-BCBD-ACTest3AB-ABPages116-1171.553.185.507.501.yes3.no5.no7.no9.yes11.noPages187-188Pages122-123Exercises(A)1.yes3.no5.yes7.no9.yes1.'1'3.'I'5.T7.T9.T11.T13.F15.FPages136-13717.T19.F21.'I'23.F25.FTest1Exercises(B)1.exterior3.corresponding5.corresponding7.base9.rightI.3603.(a)four(b)7205.1800r@jQ7.1209.14411.§!)0Test2Pages203-205Test21.F3.T5.'I'7.F9.F11.'I'13.F15.F17.'I'19.F21.T23.T25.'I'1.'1'3.'I'5.F7.'I'9.F11.F13.'I'15.Fl..&. ANSWERSTOEXERCISES451450FUNDAMENTALSOFCOLLEGEGEOMETRY(c)5/3(d)1/3(e)b/a(f)r/s5.(a)12(b)vTI69.30gal11.lUin.29.T17.T19.F21.F23.T25.T27.FPages255-256Test31.7.23.105.207.169.yes11.1513.1515.2413.51.303.1205.14in.7.13in.9.511.415.10817.36Page265Pages215-21613.36.5[t1.F3.T5.T7.F9.F11.F13.T15.F23.6025.(a)90(b)120Pages267-26917.F19.T21.1101.DF:AB=EF:AE3.DC:AE=BC:AB5.CE:BE=AC:BDPages219-2217.RP:PT=PT:PS9.PJ:HP=RJ:SH11.8!13.1815.1611.(a)65(b)651.100;140;66;543.(a)50(b)505.407.509.60Pages272-273Pages238-2411.63.lli5.207.15.39.811.1013.915.1825;mLf3=9017.819.2421.823.30.625.161.603.255.1607.509.3011.4013.mLa=70;s=7019.mLa=80;mL{3=3515.mLa=30;mLf3=6017.mLa=45;mL{3=Pages276-277Pages242-2441.103.10f5.177.12.89.13!11.2013.615.8Test117.9.619.5.521.71.perpendicular3.chords5.one7.diameterPages278-2819.supplementary11.perpendicular13.congruent----------TesfI---------.-Test2a+b1.PT3.similar5.ECXDC7.2011.613.3:71.T3.T5.F7.F9.F11.F13.F15.T9'b17.F19.F21.F23.F25.T15.8:5Test2Test31.F3.F5.T7.F9.F11.F13.F15.T17.F1.mLa=70;mL{3=80;s=603.mLa=88;mLf3=65;s=46.19.F7.mLa=17.5;mL{3=72.5;s=1105.mLa=72;mLf3=55.5;s=IIIPages246-248Test31.(a)2:3(b)5:3(c)3:5(d)2:3(e)4:53.64:3455.72;187.3:11.103.13.8565.9.7987.89.3011.2013.139.440:2111.68/83=0.81913.7T:l15.AB:AC=1.25:1;BC:CD=2.1:117.DE:BE=AE:CE=aconstantPages285-287Pages251-25211.a+c>b+d13.xx17.mLABC>mLDEF(f)5/93.(a)5/2(b)4/919.AD>BE21.BDmLA1.(a)16/5(b)6/5(c)3/2(d)4.8(e)102/7'..& .ANSWERSTOEXERCISES453452FUNDAMENTALSOFCOLLEGEGEOMETRY(b)3.4248;(c)17.04997.4:99.65.6ft211.76,986fe13.703.36in.2Page29715.123.84ft217.6949.3ft219.483.48in.I./TILB>mLA>rnLC3.NM5.STPages357-359Pages300-302Test1Test11.pi3.27.75.(radius)27.21.greater3.>5.<7.<9.>11.sum13.<15.>Test217.<19.RT1.T3.F5.T7.F9.F11.F]3.FTest2Test31.T3.T5.T7.F9.T11.F13.F15.F17.T19.T1.8.5in.23.62.4frZ5.114in.27.157ftPages337-339Pages361-362Test11.{xix>-2}3.{-2,O,2}5.{}1.perpendicularbisector3.circle5.27.37.L6---L---1L49.bisector11.circle13.twopoints'""-3-2-102Test29.tIII6III6-.-=1.(d)3.(d)5.(d)7.(c)9.(d)11.(e)13.(b)0246810Page34411.<:I.IIII4-4-202461.(a)28ft2;(b)12.5ft2;(c)77/8ft23.140yd25.307.309.10in.211.15in.2<:IIII::.II~6-4-2024Pages347-348(a)13.1.160in.23.71!-in.=7.84in.5.187in.2(approx.)7.(a)168in.2(b)168in.2(c)168in.2(d)42in.2(e)84in.2(f)84in.29.17~in.11.128in.213.36in.2<:IIII6III~6-4-2024(b)Pages350-3521.2313.1125.1877.250.29.9.611.6.413.9.24<:I6II&-4-]5.15.617.8in.19.8.48in;(b)67.]in.21.(a)72in.2;(b)1352in.2-3-2-1012(a)23.(a)21.1in.2;(b)117]in.215.(IIIIPages354-356~2-3-2-10(b)].(a)33;(b)]6.0;(c)204.63.(a)89;(b)27.88;(c)21.75855.(a)2.79;[~ 454FUNDAMENTALSOFCOLLEGEGEOMETRYANSWERSTOEXERCISES455<:11tIIII5.7.>-3-2-10123(a)17.j2~i1I]]2,11:><:-3-2-10123(b)<:IIIIIII)-4-2024(a)19.--4-2024(b)91II.Pages365-367141-13.A(l,0);B(-i,0);C(2,0);D(-t0);£(0,~);PI(2,~);P2(-i,1);P3(-t-2);I4P4(i,-~)5.(a)2;(b)3;(e)4;(d)5;(e)5;(f)9;(g)3;(h)4;(i)3;(j)2;(k)5;(I)313.51-2IXI+X2115.Y2-YI17.(0,4)19.I22-2I02461iPage3701I11-2I-22461.(a)5;(b)13;(e)17;(d)vTi3(e)812(f)1212(g)2Y683.11.4145.24I1lIIII.(a)(3,2);(e)(-Ii,-~);(b/2,a/2)-2Ii-Page3751.23.-I5.-i-7.ABIICD9.ABIICD11.AB1-CDPage37913.IS.41-I1.3.IIH-III2!46=-~-=-2--i---20246-2-2-4-4-6;,----------------------------------------------l.&.. 456FUNDAMENTALSOFCOLLEGEGEOMETRYANSWERSTOEXERCISES457Pages381-382Pages403-404I.4x-y-25=03.2x+v+4=01.21Oin.33.400inY5.972in.'!7.312in.39.1164in.25.y=-37.2x-y-1=0Pages405-4069.311.-*13.215.017.2x+y=019.x-y=O1.471in.33.1570ft.25.938in.37.405in.29.127,000ft311.15,400ft221.x-2y=0Pages408-409Page3851.no3.829in.25.5300gals7.7850fr29.206lbI.4;3;-i3.-5;none;notdefined5.5x-y-7=07.2x+y-5=0Pages410-4119.3x-y-9=011.2x-y+5=013.3x+y+4=015.x=01.904in.33.924in.25.4190ft37.3247Tft2=1020ft29.1130in.317.x=319.x+y-8=011.3iV7Tin.3=208in.321.3x-y+6=023.x+5y=0Pages413-415Pages386-387Test1Test1I.line3.V='}7TR35.anynonnegativeintegralnumber7.one9.line1.(0,0)3.fourth5.(0,0)7.309.-6i11.2ndand4th11.six13.line15.plane13.(1,-2)15.-8yTest2Test2HL£n-3--I5-.-E~~l-h---¥~l-+.--¥I+.-+19.F21.F23.F25.TI.V4253.'l5.14x-9y+12=07.15x-2y-140=09.28.3(approx.)11.3x-y+6=013.2x-y+9=OProblemsTestPages394-3961.41.25ft33.9040ft25.204in.27.2790in.31.anynonnegativeintegralnumber3.yes5.no7.no9.one11.yes13.yes15.anynonnegativeintegralnumber17.yes19.yes(seeFig.14.6)21.one23.notnecessarilv27.noconclusion29.PlaneMNint~rsectsplaneRSiniff.31.EjIIWZ33.PointQliesinplaneRS.35.i!andEKwillnotintersect,butneednotbeparallel.37.ABIIIT39.PQ1-planeKL.Pages400-4015.384in.27.2ft39.60ft211.480in.213.12,960in.315.4032in.217.400/27yd319.5.4galsiI;il..&.. IndexA.A.similaritycorollary,260subtractiontheoremof,103A.A.A.similaritytheorem,259supplementary,35Abscissa,362symmetrictheoremfor,103Absolutevalue,19transitivetheoremfor,103Acuteangle,31trihedral,390Acutetriangle,37trisectionof,312Ahmes,353vertex,129Altitude,ofcone,404vertexof,24ofcylinder,407vertical,31ofequilateraltriangle,350Arc(s),211ofparallelogram,186additionof,214ofprism,397congruent,215,223ofpyramid,401intercepted,217oftrapezoid,185major,211,213oftriangle,129,260,269,270measureof,213Analyticgeometry,330minor,211,213Angle(s),24semicircular,211acute,31,309Archimedes,2,208additiontheoremof,103Area,340adjacent,30,90,92,183ofcircle,354alternate-exterior,158ofcircularcylinder,408al~mateinterior,15&of<;"II~,404base,37,126,129ofequilateraltriangle,350bisectorof,33,105,130ofparallelogram,345central,211,213ofpolygon,341complementary,35,168postulate,342congruent,33,172,223ofprism,398constructionof,306ofpyramid,402corresponding,107,158ofrectangle,343dihedral,32,388ofrhombus,346exterior,130,131,158,167,185ofrightcircularcone,404exteriorof,25ofsphere,409inscribed,217,218ofsquare,343interior,130,158oftrapezoid,348interiorof,25oftriangle,346measureof,26A.S.A.,120obtuse,31,168,309Assumption,43reflexivetheoremfor,103Axis,360right,31,33,390polyhedral,390Base,ofcone,404sidesof,24ofcylinder,407straight,31,32ofparallelogram,186459ir,&. 60INDEX461INDEXEdge,25,288,390ofarcs,294-295ofarcs,215,223ofpolygon,184Einstein,Albert,155ofsegments,32,102ofchords,295,296ofpyramid,401Element,404ofsegments,288ffoftriangles,107,113,119,120oftrapezoid,185Ellipse,332Ioftriangle,37Conicsection,411senseof,284Conjunction,53Endpoint,20theoremof,284Between,20ofarc,157Intersection,6Construction(s),303Bisector,ofangle,33,104ofray,22Interval,21ofsegment,20,105,227,369ofanglebisector,306Equation,interceptform,383Irrationalnumber,17ofcircles,209point-slopeform,380Isosceles,36,185ofcongruentangles,306~enter,ofcircle,209slope-y-interceptform,381impossible,312:ofsphere,211two-pointform,381hords,209,227,228ofparallellines,311Lateralarea,ofcone,404angleformedby,236ofperpendicularbisector,309Equivalence,logical,65ofcylinder,407congruent,223ofperpendicularline,307,310Euclid,2,155,208ofprism,397ircle(s),206,209Contraposition,67Excludedmiddle,lawof,62ofpyramid,402Converse,64Existence,147Legsofrighttriangle,37arcof,211Coordinate,17,362Extremes,248Linearpair,32areaof,354centerof,209,231Coordinategeometry,360Line(s),12Coplanar,12Faces,ofdihedralangle,390construction,305centralangleof,211Corollary,84ofprism,397curved,12chordof,209circumferenceof,352Correspondence,one-to-one,17,107ofpyramid,401equationsof,380-383Correspondingparts,101,107,125Field,propertiesof,73oblique,148circumscribed,210,314Figures,congruent,33,101,113-132parallel,13,139,154-163Cube,398concentric,209Footofperpendicular,34Cylinder,407perpendicular,33,91,92,147,149congruent,209altitudeof,407segmentof,20diameterof,209Generatrix,404,406sideof,25baseof,407exteriorof,210Geometry,1skew,139great,212circular,407coordinate,360straight,12inscribed,314directrixof,406Euclidean,155,208Lobachevsky,Nicholas,155elementof,406interiorof,210non-Euclidean,155Locus(1oci),319-335generatrixof,406radiusof,209space,388intersectionof,325lateralareaof,407tangentto,210,231-238Graphing,360,377theoremsof,322-325rightcircular,4072ircumference,352Greekalphabet,419Logic,51surfaceof,40620llinear,12totalareaof,40820mpass,209,303Half-line,21Measurement,41,213,25320mplement,35volumeof,408Half-plane,25Median,oftrapezoid,18520nclusion,41,82clos~d,25oftriangle,130Decagon,18420ne,404edgeof,25Midpoint,20,198,199Deduction,43altitudeof,404Hemisphere,212formula,369Definitions,circular,11baseof,404Hexagon,184Modusponens,60Degree,angle,26circular,405Historicalnotes,1,2,150,155,206,303,312,Modustollens,61Denyingthealternative,62directrixof,404353generatrixof,404Diagonal,184,189,192Hyperbola,334Negation,56,57lateralareaof,404Diameter,209Hypotenuse,37,175Numbers,abstract,246rightcircular,405Directrix,404,406'Hypothesis,82irrational,17slantheightof,405Disjointsets,7IDistance,formula,367rational,17surfaceof,405Implication,58real,16vertexof,404frompointtoline,34,227,228Inclination,371,372volumeof,405betweentwopoints,18Induction,43Octagon,184ongruence,101Doublingthecube,312Inequalities,283Order,18,283ofangles,33,103,172,223Drawing,303ofangles,287-290Ordinate,362r~ ..,INDEX46362INDEXregular,397diagonalsof,188Square,186,343,349rigin,362Region,areaof,341Squareroottable,421right,397polygonal,340Squaringthecircle,312totalareaof,397Parabola,328Relativity,theoryof,155S.S.S.,132Parallelepiped,397volumeof,399Proof,81Rhombus,186,189Statement,51rectangular,397Rieman,Bernard,155Straight,12formal,81Parallellines,13,139,153-163,188,253-Rigidity,110edge,303geometric,116255,373indirect,142Rulerpostulate,79Subset,6constructionof,311bymeasurement,41Supplement,35planes,140,153Proportion,248S.A.S.,113Surface,12postulateof,155extremesof,248Secant,210,236,274,275conical,404Parallelogram,186-192Segments,20cylindrical,406altitudeof,186meansof,248additiontheoremsfor,102spherical,409theoremsof,249-251areaof,345Proportional,fourth,249bisectortheoremfor,104diagonalsof,189,192endpointof,20Tangents,toacircle,210,230-238mean,249~entagon,184measureof,20fromexternalpoint,232Iequiangular,184segments,261midpointof,20,105,227Thales,2,208equilateral,184Protractor,27reflexivetheoremfor,102Theorem,43,75,82Pyramid,401externalangleof,185ofsecant,274Transversal,158regular,184altitudeof,401subtractiontheoremfor,102Trapezoid,185Perimeter,184baseof,401symmetrictheoremfor,102altitudeof,185Perpendicular,33,91,92,147,150,153,157,lateralareaof,402transitivetheoremfor,102areaof,348230lateraledgesof,401Semicircle,211,218basesof,185Pi(1r),353lateralfacesof,401angleinscribedin,218isosceles,185Plane(s),12regular,401Set,2legsof,185intersecting,14slantheightof,401complementof,6medianof,185parallel,140,141vertexof,401elementsof,3Triangle(s),36perpendicular,141,393volumeof,402empty,3acute,37lineperpendicularto,391Pythagoras,2,208equal,5altitudeof,129Plato,2Pythagoreantheorem,270finite,3"TI'"of.141>Playfair,155null,3baseof,37Point,11Quadrant,363Quadrilateral,184universal,5congruent,111,113,114,119,120,131collinear,12equiangular,37,173Similarpolygons,256-259Polygon(s),183Similartriangles,259-269equilateral,37,173,350areaof,341Radius,209Ratio,245Slantheight,401,405exteriorangleof,130,167convex,183Slope,371exteriorof,36diagonalof,184ofsimilitude,258Rationalnumbers,17Solid,12interiorof,36inscribed,210Ray,22Sphere(s),211isosceles,36regular,184directionof,196areaof,409labelingof,36similar,256Polyhedralangles,388endpointof,22centerof,211medianof,130opposite,22concentric,211obtuse,37edgesof,390Realnumbers,72congruent,211right,37,173,175facesof,390equalityof,72exteriorof,211rigidityof,110verticesof,388Polyhedron,396orderpropertiesof,73greatcircleof,212scalene,36Prism,396Reasoning,converse,64interiorof,211similar,259-269deductive,43planetangentto,211vertexangleof,36altitudeof,397inductive,43propertiesof,212Trisectionofangle,312areaof,397logical,51radiusof,211Truth,44basesof,396smallcircleof,212value,51lateralareaof,397Rectangle,areaof,343lateraledgesof,397definitionof,186,188volumeof,409l~ 464INDEXUnion,7ofisoscelestriangle,36Uniqueness,147ofpolygon,183Unitofmeasurement,212ofpyramid,401Universalset,5Verticalangles,31,91Volume,ofcircularcone,405Validity,44ofcircularcylinder,408Venndiagram,5ofprism,399Vertex,ofangle,24ofpyramid,402ofcone,404ofsphere,409,..