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ProofoftheAngleSumPropertyofTriangles. Thefollowinggivesaprooftheanglesumpropertyoftriangles,alongwiththeproofsofthe theoremsusedintermediately.Someofthesehavebeentakenbacktoaxioms,postulatesand definitionsandafewhavebeenleftassuggestedexercises. 1.Toprove:Sumoftheanglesofatriangleis2rightangles. SideBCisproducedtoD.(Euclid’s2ndPostulate) (1.1) Givenalineandapointnotonit,thereexistsalineparalleltothegivenlinethroughthegiven point. (1.2) Given(1.2) ThereexistsastraightlineCEparalleltoAB. (1.3) ACisastraightlinethatfallsonCEandAB. (1.4) Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother (1.5) Given1.3,1.4and1.5, <ACE=<BAC BDisastraightlinethatfallsonCEandAB. (1.6) (1.7) Astraightlinefallingonparallelstraightlinesmakesapairofcorrespondinganglesequal. (1.8) (1.9) <ABC+<ACB+<<BCA=<ECD+<ACE+<BCA (1.10) Given1.1,<ECD+<ACE+<BCAistworightangles. (1.11) Given1.10and1.11,thesumoftheanglesofatriangleis2rightangles. (1.12) Given1.3,1.7and1.8 <ECD=<ABC Given1.6and1.9, Thesumoftheanglesinthetriangle= NotethatthisproofappliestoANYtriangle,notjustthesampleoftrianglesthatwehappentohave measured.Thisistheessenceofamathematicalproof. InthisStatementsinbold1.2,1.5,1.8and1.11needtobeproved.Startingwith1.5. 2(1.5).ToProve:Astraightlinefallingonparallelstraightlinesmakesthe alternateanglesequaltooneanother. LetABandCDbeapairofparallellines,andFisastraightlinethatfallsonthem. (2.1) <AGHand<GHDareapairofalternateangles. (2.2) Contrarytowhatwewishtoprove,Let<AGHnotequalto<GHD. (2.3) Let<GHDbethegreaterofthetwo. (2.4) Add<CHGtoboth. (2.5) (2.6) Given2.4and2.5 <GHD+<CHGisgreaterthan<AGH+<CHG Ifastraightlinestandsonastraightline,thenitmakeseithertworightanglesorangleswhose sumequalstworightangles. (2.7) <GHD+<CHGis2rightangles. Given2.6and2.8,<AGH+<CHGislessthan2rightangles (2.8) (2.9) Ifastraightlinefallingontwostraightlinesmakestheinterioranglesonthesamesidelessthan tworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhichare theangleslessthanthetworightangles. (2.10) Given2.9and2.10 LinesABandCDmeetonthesideofAandC. (2.11) Thiscontradicts1.Thereforeassumption3isnottrue. Itfollowsfrom2.11that<AGHISequalto<GHD HenceProved. (2.12) Here2.7needstobeproved.2.10isthefifthpostulate. 3(1.8)ToProve:Astraightlinefallingonparallelstraightlinesmakesthe correspondinganglesequaltooneanother. LetABandCDbeapairofparallellines,andFisastraightlinethatfallsonthem. (3.1) <EGBand<GHDareapairofcorrespondingangles. (3.2) Whenapairoflinesintersect,apairofverticallyoppositeanglesarecongruent. (3.3) Given3.1and3.3,<AGH=<EGB (3.4) Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother.(3.5) Given3.5,<AGH=<GHD (3.6) (3.7) Given3.4and3.6, <EGB=<GHD Henceproved. Statement3.5nowisanalreadyprovedtheorem,whichcanbeused.Statement3.3needstobe proved. 4(1.2)ToProve:Givenalineandapointnotonit,thereexistsalineparallelto thegivenlinethroughthegivenpoint. BCisalineandAisapointthatisnotonit. (4.1) TakeapointDonBCandjoinAD. (4.2) Construct<DAEsuchthatitisequalto<ADC. (4.3) {Thatthisispossiblecanbeproved–Euclidproposition23} ExtendEAtoEF.{2ndPostulate} (4.4) Ifastraightlinesfallsonapairoflinessuchthatapairofalternateinterioranglesarecongruent, thenthepairoflinesareparallel. (4.5) Given4.2,4.3,4.4and4.5, EFisparalleltoBC. Henceproved. Statement4.5,whichistheconverseofTheorem2provedaboveneedstobeproved. Weproved1.2,1.5and1.8ofproof1above.Intheprocessweseethatwehaveusedafewmore unprovedresults. Wewillprove2ofthese– Whenapairoflinesintersect,apairofverticallyoppositeanglesarecongruent. Ifastraightlinesfallsonapairoflinessuchthatapairofalternateinterioranglesarecongruent, thenthepairoflinesareparallel. 5(3.3)ToProve:Whenapairoflinesintersect,apairofverticallyopposite anglesarecongruent. Ifastraightlinestandsonastraightline,thenitmakeseithertworightanglesorangleswhose sumequalstworightangles. (5.1) AEisastraightlinethatstandsonCD. (5.2) Given(5.1)and(5.2) <AEC+<AED=2rightangles. (5.3) Similarly,CEstandsonAB. (5.4) (5.5) Given(5.1)and(5.4) <AEC+<CEB=2rightangles Given(5.3)and(5.5) <AED=<CEB Similarlyitcanbeshownthat<AEC=<DEB (5.6) (5.7) HenceProved. 6.(4.5)ToProve:Ifastraightlinesfallsonapairoflinessuchthatapairof alternateinterioranglesarecongruent,thenthepairoflinesareparallel. ABandCDareapairoflinesandEFfallsonthemsuchthat<BEF=<CFE. (6.1) (6.2) WeneedtoprovethatABisparalleltoCD Contrarytowhatweneedtoprove,assume,LetABnotparalleltoCD. If(6.2)istrue,ABandCDmeetwhenextended.LetthemmeetonthesideofBandDatGsay.(6.3’) E,F,Gformtheverticesofatriangle. (6.3) (6.4) Given6.1, <AEFanexteriorangleofatriangleCFE=<FEGaninterioroppositeangle. But,Inanytriangle,ifoneofthesidesisproduced,thentheexteriorangleisgreaterthaneitherof theinteriorandoppositeangles.” (6.5) (6.4)contradicts(6.5) Thereforeassumption6.2cannotbetrue. ThereforeADisparalleltoCD. Henceproved. Afewquestions 1) Inproof6,wehaveyetanotherunprovedresultnamely-“Inanytriangle,ifoneofthe sidesisproduced,thentheexteriorangleisgreaterthaneitheroftheinteriorand oppositeangles.” Isthefollowinganacceptableproofforthisresult?Whyorwhynot? Proof: Thesumoftheanglesofatriangleistworightangles. (1) Given1,<ABC+<BCA+<CAB=2rightangles. (2) Ifastraightlinestandsonastraightline,thenitmakeseithertworightanglesorangles whosesumequalstworightangles. (3) Given2,<BCA+<ACD=2rightangles. (4) (5) (6) Given2and4 <ABC+<CAB=<ACD <ABCand<CABarebothpositivequantitiesgivenABCisatriangle Given5and6,<ACDisgreaterthanboth<ABCand<CAB.Henceproved. 2) Furtherexploration: Youcouldtrytoprovethefollowingstatementswhichhavebeenusedintheproof. Ifastraightlinestandsonastraightline,thenitmakeseithertworightanglesorangles whosesumequalstworightangles.” Itispossibletoconstructanangleequaltoagivenangleonagivenstraightlineandapoint onit. Inanytriangle,ifoneofthesidesisproduced,thentheexteriorangleisgreaterthaneither oftheinteriorandoppositeangles.” Intheprocess,youmayuseafewmoretheoremswhichrequiretobeprovedoryoumay useonlyaxioms,definitionsandpostulatesthatareacceptedastrue.