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24 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)? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org https://flic.kr/p/3GZScG CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 CC BY Anders Sandberg SECONDARY MATH I // MODULE 7 25 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 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,itcontainsfourrightanglesand allsidesarecongruent 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). Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 26 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Congruent Triangles CriteriaUsed (ASA,SAS,SSS) IfIsayΔRST≅ΔXYZ basedonSSS HowIknowthesidesand/orangles requiredbythecriteriaarecongruent 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. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 7. 5 Congruent Triangles to the Rescue – Teacher Notes A Practice Understanding Task Purpose:Thepurposeofthistaskistoprovidestudentswithpracticeinidentifyingthecriteria theymightuse—ASA,SASorSSS—todetermineiftwotrianglesembeddedinanothergeometric figurearecongruent,andthentousethosecongruenttrianglestomakeotherobservationsabout thegeometricfiguresbasedontheconceptthatcorrespondingpartsofcongruenttrianglesare congruent.Asecondarypurposeofthistaskistoallowstudentstocontinuetoexaminewhatit meanstomakeanargumentbasedonthedefinitionsoftransformations,aswellasbasedon propertiesofcongruenttriangles.Thefocusshouldbeonusingcongruenttrianglesand transformationstoidentifyotherthingsthatcanbesaidaboutageometricfigure,ratherthanonthe specificpropertiesoftrianglesorquadrilateralsthatarebeingobserved.Theseobservationswillbe moreformallyprovedinSecondaryII.Theobservationsinthistaskalsoprovidesupportforthe geometricconstructionsthatareexploredinthenexttask. CoreStandardsFocus: G.CO.7Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesare congruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesare congruent. G.CO.8Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthe definitionofcongruenceintermsofrigidmotions. SeealsoMathematicsInoteforG.CO.6,G.CO.7,G.CO.8:Rigidmotionsareatthefoundationofthe definitionofcongruence.Studentsreasonfromthebasicpropertiesofrigidmotions(thatthey preservedistanceandangle),whichareassumedwithoutproof.Rigidmotionsandtheirassumed propertiescanbeusedtoestablishtheusualtrianglecongruencecriteria,whichcanthenbeusedto proveothertheorems. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 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 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 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 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 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 27 SECONDARY MATH I // MODULE 7 7.5 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 READY, SET, GO! Name PeriodDate READY Topic:Transformationsoflines,connectinggeometryandalgebra. Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichis thepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethe transformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline. 1. 2. N' M' N M a.DescriptionofTransformation: b.Equationforpre-image: c.Equationforimage: Useforproblems3thorugh5. a.DescriptionofTransformation: b.Equationforpre-image: c.Equationforimage: 3.a.DescriptionofTransformation: b.Equationforpre-image: c.Equationforimage: 4.Writeanequationforalinewiththesameslope thatgoesthroughtheorigin. 5.Writetheequationofalineperpendicularto theseandthoughthepointO’. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 28 SECONDARY MATH I // MODULE 7 7.5 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgood timingtoconsidersimilarworkwithtables. 6.Matchthetableofvaluesbelowwiththeproperfunctionrule. I II x -1 0 1 2 f(x) 16 14 12 10 III x -1 0 1 2 A.! ! = −! ! − ! + ! B.! ! = −! ! − ! + !" C.! ! = −! ! − ! + ! f(x) 14 12 10 8 IV x -1 0 1 2 f(x) 12 10 8 6 V x -1 0 1 2 D.! ! = −! ! + ! + ! E.! ! = −! ! + ! + !" f(x) 10 8 6 4 x -1 0 1 2 SET Topic:UseTriangleCongruenceCriteriatojustifyconjectures. Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsa conjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsand conjecturestatementcreateanargumentthatjustifiestheconjecture. 7.Truestatements: Conjecture: ∠A ≅ ∠C PointMisthemidpointof!" ∠!"# ≅ ∠!"# a.Istheconjecturecorrect? !" ≅ !" b.Argumenttoproveyouareright: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org f(x) 8 6 4 2 29 SECONDARY MATH I // MODULE 7 7.5 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 8.Truestatements ∠ !"# ≅ ∠ !"# !" ≅ !" 9.Truestatements ∆ !"#isa180° rotationof∆ !"# Conjecture:!"bisects∠ !"# a.Istheconjecturecorrect? b.Argumenttoproveyouareright: Conjecture:∆ !"# ≅ ∆!"# a.Istheconjecturecorrect? b.Argumenttoproveyouareright: GO Topic:Constructionswithcompassandstraightedge. 10.Whydoweuseageometriccompasswhendoingconstructionsingeometry? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 30 SECONDARY MATH I // MODULE 7 CONGRUENCE, CONSTRUCTION AND PROOF- 7.5 Performtheindicatedconstructionsusingacompassandstraightedge. 11.Constructarhombus,usesegmentABasonesideandangleAasoneoftheangles. 12.ConstructalineparalleltolinePR andthroughthepointN. 13.ConstructanequilateraltrianglewithsegmentRSasoneside. 14.Constructaregularhexagoninscribed inthecircleprovided. 15.ConstructaparallelogramusingCDasoneside andCEastheotherside. 16.BisectthelinesegmentLM. 17.BisecttheangelRST. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 7.5