SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
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Lesson 9 Claims and Conjectures
A Solidify Understanding Task
ThediagramfromHowDoYouKnowThat?hasbeenextendedbyrepeatedlyrotatingtheimage
trianglesaroundthemidpointsoftheirsidestoformatessellationoftheplane,asshownbelow.
Usingthisdiagram,wewillmakesomeconjecturesaboutlines,anglesandtrianglesandthenwrite
proofstoconvinceourselvesthatourconjecturesarealwaystrue.
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
VerticalAngles
Whentwolinesintersect,theoppositeanglesformedatthepointofintersectionarecalledvertical
angles.Inthediagrambelow,∠1and∠3formapairofverticalangles,and∠2and∠4form
anotherpairofverticalangles.
Examinethetessellationdiagramabove,lookingforplaces
whereverticalanglesoccur.(Youmayhavetoignoresome
linesegmentsandanglesinordertofocusonpairsofvertical
angles.Thisisaskillwehavetodevelopwhentryingtoseespecificimagesingeometricdiagrams.)
Basedonseveralexamplesofverticalanglesinthediagram,writeaconjectureaboutverticalangles.
Myconjecture:
ExteriorAnglesofaTriangle
Whenasideofatriangleisextended,asinthediagrambelow,theangleformedontheexteriorofthe
triangleiscalledanexteriorangle.Thetwoanglesofthetrianglethatarenotadjacenttotheexterior
anglearereferredtoastheremoteinteriorangles.Inthediagram,∠4isanexteriorangle,and∠1
and∠2arethetworemoteinterioranglesforthisexteriorangle
Examinethetessellationdiagramabove,lookingforplaces
whereexterioranglesofatriangleoccur.(Again,youmayhave
toignoresomelinesegmentsandanglesinordertofocuson
trianglesandtheirverticalangles.)
Basedonseveralexamplesofexterioranglesoftrianglesinthediagram,writeaconjectureabout
exteriorangles.
Myconjecture:
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
ParallelLinesCutByaTransversal
Whenalineintersectstwoormoreotherlines,thelineiscalledatransversalline.Whentheother
linesareparalleltoeachother,somespecialanglerelationshipsareformed.Toidentifythese
relationships,wegivenamestoparticularpairsofanglesformedwhenlinesarecrossed(orcut)bya
transversal.Inthediagrambelow,∠1and∠5arecalledcorrespondingangles,∠3and∠6are
calledalternateinteriorangles,and∠3and∠5arecalledsamesideinteriorangles.
Examinethetessellationdiagramabove,lookingfor
placeswhereparallellinesarecrossedbyatransversal
line.
Basedonseveralexamplesofparallellinesand
transversalsinthediagram,writesomeconjectures
aboutcorrespondingangles,alternateinterioranglesand
samesideinteriorangles.
Myconjectures:
JustifyingOurConjectures
Inthenexttaskyouwillbeaskedtowriteaproofthatwillconvinceyouandothersthateachofthe
conjecturesyouwroteaboveisalwaystrue.Youwillbeabletouseideasabouttransformations,
linearpairs,congruenttrianglecriteria,etc.tosupportyourarguments.Agoodwaytostartisto
writedowneverythingyouknowaboutthediagram,andthenidentifywhichstatementsyoumight
usetomakeyourcase.Togetreadyforthenexttask,revisiteachoftheconjecturesyouwroteabout
andrecordsomeideasthatseemhelpfulinprovingthattheconjectureistrue.
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SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
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READY Topic:PropertiesofQuadrilaterals
1. Usewhatyouknowabouttrianglestowriteaparagraphproofthatprovesthatthesumofthe
anglesinaquadrilateralis360º.
2. FindthemeasureofxinquadrilateralABGC.
Matchtheequationwiththecorrectlineinthegraphoflinesp,q,r,ands.
3. ! = !! ! + 2
4. ! = − !! ! + 2
5. ! = !! ! + 4
6. ! = − !! ! + 4
7. Describetheshapemadebytheintersection ofthe
4lines.Listasmanyobservationsasyou canaboutthe
shapeanditsfeatures.
READY, SET, GO! Name Period Date
r
pq
s
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Lesson 9
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
SET Topic:Parallellinescutbytransversal,verticalanglesandexteriorangleofatriangleLabeleachpictureasshowingparallellineswithatransversal,verticalangles,oranexteriorangleofatriangle.Highlightthegeometricfeatureyouidentified.Canyoufindall3featuresin1picture?Where?
8. 9. 10.
11. 12. 13.
14. 15. 16.
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Lesson 9
SECONDARY MATH II // MODULE 5
GEOMETRIC FIGURES – 5.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
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Findthevalueofthe2remoteinterioranglesinthefiguresbelow.
17. 18. 19.
Indicatewhethereachpairofanglesiscongruentorsupplementarybytrustinghowtheylook.Linespandqareparallel.
20. ∠5 !"# ∠8
21. ∠2 !"# ∠6
22. ∠2 !"# ∠8
23. ∠4 !"# ∠6
24. ∠3 !"# ∠5
25. ∠1 !"# ∠3
GO Topic:Complementaryandsupplementaryangles
Findthecomplementandthesupplementofthegivenangles.Itispossibleforthecomplementorsupplementnottoexist.
26. 37° 27. 59° 28. 89°
29. 111° 30. 3° 31. 90°
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Lesson 9