7. 5 congruent triangles to the rescue - utah education …€¦ ·  · 2017-10-26part 1 zac and...

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

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7. 5 Congruent Triangles

to the Rescue

A Practice Understanding Task

Part1

ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent:

Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertex

pointoftheanglemadeupbythetwocongruentsides,andthemidpointofthethirdside.

Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.

Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.

Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.

Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.

1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineof

symmetry?Ifso,whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabout

hisstatement?

2. WhatelsewouldZacneedtoknowaboutthecreaselinethroughinordertoknowthatitisa

lineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)

3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)

createstwocongruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASor

SSS—couldheusetosupportthisclaim?Describethesidesand/oranglesyouthinkare

congruent,andexplainhowyouknowtheyarecongruent.

4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes

thatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenot

formedbythetwocongruentsides)?

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes

thatimplyaboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthis

line—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non-congruent

sideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthe

vertexangleformedbythetwocongruentsides.)

Part2

LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotational

symmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinitionyou

madesomeobservationsaboutsides,angles,anddiagonalsofvarioustypesofquadrilateralsbased

onyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbe

furtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZac

andSionedidintheirworkwithisoscelestriangles.

Pickoneofthefollowingquadrilateralstoexplore:

• Arectangleisaquadrilateralthatcontainsfourrightangles.

• Arhombusisaquadrilateralinwhichallsidesarecongruent.

• Asquareisbotharectangleandarhombus,thatis,itcontainsfourrightanglesandallsidesarecongruent

1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthe

quadrilateralA,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.

2. Basedon(1)yourdrawing,(2)thegivendefinitionofyourquadrilateral,and(3)information

aboutsidesandanglesthatyoucangatherbasedonlinesofreflectionandrotational

symmetry,listasmanypairsofcongruenttrianglesasyoucanfind.

3. Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused—ASA,SASorSSS—to

determinethatthetwotrianglesarecongruent,andexplainhowyouknowthattheangles

and/orsidesrequiredbythecriteriaarecongruent(seethefollowingchart).

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

CongruentTriangles

CriteriaUsed(ASA,SAS,SSS)

HowIknowthesidesand/oranglesrequiredbythecriteriaarecongruent

IfIsayΔRST≅ΔXYZ

basedonSSS

thenIneedtoexplain:

• howIknowthat

�

RS ≅ XY ,and• howIknowthat

�

ST ≅ YZ ,and• howIknowthat

�

TR ≅ ZX soIcanuseSSScriteriatosayΔRST≅ΔXYZ

4. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethe

congruenttrianglestojustifysomethingelseaboutthequadrilateral,suchas:

• thediagonalsbisecteachother

• thediagonalsarecongruent

• thediagonalsareperpendiculartoeachother

• thediagonalsbisecttheanglesofthequadrilateral

Pickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandtryto

writeanargumentthatwouldconvinceZacandSionethatthestatementistrue.

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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7.5

READY Topic:Transformationsoflines,connectinggeometryandalgebra.Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichisthepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethetransformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline.1.

2.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:Useforproblems3thorugh5.

3.a.DescriptionofTransformation:b.Equationforpre-image:c.Equationforimage:4.Writeanequationforalinewiththesameslopethatgoesthroughtheorigin.5.WritetheequationofalineperpendiculartotheseandthoughthepointO’.

M

N

M'

N'

READY, SET, GO! Name PeriodDate

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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7.5

Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgoodtimingtoconsidersimilarworkwithtables.6.Matchthetableofvaluesbelowwiththeproperfunctionrule.I II III IV V

x f(x)-1 160 141 122 10

x f(x)-1 140 121 102 8

x f(x)-1 120 101 82 6

x f(x)-1 100 81 62 4

x f(x)-1 80 61 42 2

A.! ! = −! ! − ! + ! D.! ! = −! ! + ! + ! B.! ! = −! ! − ! + !" E.! ! = −! ! + ! + !" C.! ! = −! ! − ! + ! SET Topic:UseTriangleCongruenceCriteriatojustifyconjectures.Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsaconjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsandconjecturestatementcreateanargumentthatjustifiestheconjecture.

7.Truestatements: PointMisthemidpointof!"∠!"# ≅ ∠!"#!" ≅ !"

Conjecture:∠A ≅∠C a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

31

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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7.5

8.Truestatements∠ !"# ≅ ∠ !"#!" ≅ !"

Conjecture:!"bisects∠ !"#a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

9.Truestatements∆ !"#isa180°rotationof∆ !"#

Conjecture:∆ !"# ≅ ∆!"#a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

GO Topic:Constructionswithcompassandstraightedge.10.Whydoweuseageometriccompasswhendoingconstructionsingeometry?

32

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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7.5

Performtheindicatedconstructionsusingacompassandstraightedge.11.Constructarhombus,usesegmentABasonesideandangleAasoneoftheangles.12.ConstructalineparalleltolinePRandthroughthepointN.13.ConstructanequilateraltrianglewithsegmentRSasoneside.14.Constructaregularhexagoninscribedinthecircleprovided.15.ConstructaparallelogramusingCDasonesideandCEastheotherside.16.BisectthelinesegmentLM. 17.BisecttheangelRST.

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