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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Needhelp?Visitwww.rsgsupport.org

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