7. 5 congruent triangles to the rescue - utah education …€¦ · · 2017-10-267. 5 congruent...

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.5

Mathematics Vision Project

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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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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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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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7. 5 Congruent Triangles to the Rescue – Teacher Notes A Practice Understanding Task

Purpose:Thepurposeofthistaskistoprovidestudentswithpracticeinidentifyingthecriteriatheymightuse—ASA,SASorSSS—todetermineiftwotrianglesembeddedinanothergeometricfigurearecongruent,andthentousethosecongruenttrianglestomakeotherobservationsaboutthegeometricfiguresbasedontheconceptthatcorrespondingpartsofcongruenttrianglesarecongruent.Asecondarypurposeofthistaskistoallowstudentstocontinuetoexaminewhatitmeanstomakeanargumentbasedonthedefinitionsoftransformations,aswellasbasedonpropertiesofcongruenttriangles.Thefocusshouldbeonusingcongruenttrianglesandtransformationstoidentifyotherthingsthatcanbesaidaboutageometricfigure,ratherthanonthespecificpropertiesoftrianglesorquadrilateralsthatarebeingobserved.TheseobservationswillbemoreformallyprovedinSecondaryII.Theobservationsinthistaskalsoprovidesupportforthegeometricconstructionsthatareexploredinthenexttask.CoreStandardsFocus:G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.G.CO.8Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionofcongruenceintermsofrigidmotions.SeealsoMathematicsInoteforG.CO.6,G.CO.7,G.CO.8:Rigidmotionsareatthefoundationofthedefinitionofcongruence.Studentsreasonfromthebasicpropertiesofrigidmotions(thattheypreservedistanceandangle),whichareassumedwithoutproof.Rigidmotionsandtheirassumedpropertiescanbeusedtoestablishtheusualtrianglecongruencecriteria,whichcanthenbeusedtoproveothertheorems.






RelatedStandards:G.CO.10

StandardsforMathematicalPracticeofFocusintheTask:

SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP7–Lookforandmakeuseofstructure

AdditionalResourcesforTeachers:

Acopyoftheimagesusedinthistaskcanbefoundattheendofthissetofteachernotes.These

imagescanbeprintedforusewithstudentswhomaybeaccessingthetaskonacomputerortablet.

TheTeachingCycle:

Launch(WholeClass):

Makesurethatstudentsknowthedefinitionofanisoscelestriangleandgivethemseveralisosceles

trianglestofold—essentiallyrecreatingZac’spaper-foldingexperimentasdescribedinpart1ofthe

task(seeattachedhandoutofisoscelestriangles).Askstudentsiftheyseeanycongruenttriangles

insideofthefoldedisoscelestriangle,andwhatcriteriaforcongruenttriangles—ASA,SASorSSS—

theycouldusetoconvincethemselvesthattheseinteriortrianglesarecongruent.Workthroughthe

additionalquestionsinpart1withtheclass,givingstudentstimetothinkabouteachquestion

individuallyorwithapartner.

HelpstudentsseethedifferencebetweenverifyingZac’sclaim(“everyisoscelestrianglehasalineof

symmetrythatpassesthroughthevertexpointoftheanglemadeupofthetwocongruentsides,and

themidpointofthethirdside”)throughexperimentation—paperfolding—andajustificationbased

ontransformationsandcongruenttrianglecriteria.Itappearsfromfoldingonesideoftheisosceles

triangleontotheotherthattwocongruenttrianglesareformed.ThiscanbejustifiedusingtheSSS

trianglecongruencecriterion:thelinethroughthevertexandthemidpointoftheoppositesideis

commontobothinteriortriangles(S1);themidpointoftheoppositesideformstwocorresponding

congruentsegmentsintheinteriortriangles(S2);andbydefinitionofanisoscelestriangletheother

pairofsidesintheinteriortrianglesarecongruent(S3).Sincetheinteriortrianglesarecongruent






bySSS,wecanalsoconcludethatthethreecorrespondinganglesarecongruent.Thisleadstosuch

additionalpropertiesas:thebaseanglesoftheisoscelestrianglearecongruent;thevertexangleis

bisectedbythelinethroughthevertexandmidpointoftheoppositeside;andthelinethroughthe

vertexandmidpointoftheoppositesideisperpendiculartothebasesincetheanglesformedare

congruentandtogetherformastraightangle.Collectively,thesestatementsjustifyZac’sclaimthat

everyisoscelestrianglehasalineofsymmetry.

Explore(SmallGroup):

Theguideddiscussionofpart1ofthistaskwillpreparestudentstoworkmoreindependentlyon

part2.Youmaywanttoassigndifferentgroupstoaparticularquadrilateral,soallofthe

quadrilateralsgetexplored.Centertheexplorationtimeonpart2,questions2and3—lookingfor

congruenttriangles,andlistingthecriteriathatwasusedtoclaimthatthetrianglesarecongruent.

Fastfinisherscanworkonpart2,question4—justifyingotherpropertiesofquadrilateralsbasedon

correspondingpartsofcongruenttriangles.

Discuss(WholeClass):

Thefocusofthediscussionshouldbeonpart2,question2—identifyingcongruenttrianglesformed

indifferenttypesofquadrilateralsbydrawinginthediagonals.Asstudentsclaimtwotrianglesare

congruent,askthemtoexplainthetrianglecongruencecriteria—ASA,SASorSSS—theyusedto

justifytheirclaim.Astimeallows,discusssomeoftheotherclaimsthatcanbemadeaboutthe

quadrilateralsbasedoncorrespondingpartsofcongruenttriangles.

AlignedReady,Set,Go:Congruence,ConstructionandProof7.5




Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

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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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:

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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?

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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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7. 5 congruent triangles to the rescue - utah education …€¦ · · 2017-10-267. 5 congruent...

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