Download Proof of the Angle Sum Property of Triangles

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.