7. 5 congruent triangles to the rescue - utah education …€¦ · · 2017-10-267. 5 congruent...
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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
Licensed under the Creative Commons Attribution CC BY 4.0
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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
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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.
SECONDARY MATH I // MODULE 7
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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
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
Mathematics Vision Project
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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
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
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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:
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.5
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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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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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