chapter 4 euclidean geometry：4章欧氏几何

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1、Chapter4EuclideanGeometry4.1EuclideanParallelism,ExistenceofRectangles4.2ParallelismandTrapezoids:ParallelProjection4.3SimilarTriangles,PythagoreanTheorem,Trigonometry4.5TheCircleTheoremsHomework6114.1EuclideanParallelism,ExistenceofRectanglesWebeginwithadefinitionofparallellines:TwodistinctlinesLa

2、ndMaresaidtobeparallelifftheyareinthesameplaneanddonotmeet.Wealsoneedtheconceptofatransversal.Atransversalisalinethatintersectstwootherdistinctlines.Forexample,thethirdsideofatriangleisatransversal.Theauthorhasapictorialglossaryforalternateinteriorangles,correspondingangles,andinterioranglesonthesa

3、mesideofatransversal.Thesetermsapplytotheangleswhetherornotthelinesareparallel.4.1.1ParallelisminAbsoluteGeometrypage212Iftwolinesinthesameplanearecutbyatransversalsothatapairofalternateinterioranglesarecongruent,thenthelinesareparallel.ThistheoremwillbeprovedBEFOREweaccepttheaxiomofparallellines.I

4、tshowsthatparallellinesexistinAbsoluteGeometry.Proof:LetLandMbetwolinescutbyatransversalT.Letangles1and2bealternateinteriorangles.SupposeLandMarenotparallelandtheyintersectatsomepointR.Thenwehaveatriangleandoneofthealternateinterioranglesisactuallyanexteriorangleofthistriangle.Sincetheotherangleisi

5、nterior,itisactuallyaremoteinteriorangleandbytheExteriorAngleInequalitytheexteriorangleisstrictlylargerthantheremoteinteriorangle.Thiscontradictsourhypothesissooursuppositionisinerrorandthelinesareactuallyparallel.AxiomP1EuclideanParallelPostulateIfLisanylineandPanypointnotonL,thenthereexistsinthep

6、laneofLandPoneandonlyonelinethatpassesthroughPandisparalleltoL.Withthescenariointheaxiom,therearethreepossibilitiesforparallellines:·none(we’veseenthiswithSphericalGeometry)·one(thisisthefamiliarEuclideanGeometrysituation)·many(we’llseethisinChapter6withHyperbolicGeometry)Thereareimmediateandfamili

7、artheoremsandcorollariesthatfollowfrommakingthischoice.11Theorem4.1.1page214TheTransversalTheoremsforParallelismpages216&217Example2page216istheproofofTheorem4.1.3ExteriorAngleTheoremforEuclideanGeometry.Be