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
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)?
CC
BY
And
ers
Sand
berg
http
s://f
lic.k
r/p/
3GZ
ScG
27
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
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).
28
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.
29
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
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
30
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
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
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.5
Performtheindicatedconstructionsusingacompassandstraightedge.11.Constructarhombus,usesegmentABasonesideandangleAasoneoftheangles.12.ConstructalineparalleltolinePRandthroughthepointN.13.ConstructanequilateraltrianglewithsegmentRSasoneside.14.Constructaregularhexagoninscribedinthecircleprovided.15.ConstructaparallelogramusingCDasonesideandCEastheotherside.16.BisectthelinesegmentLM. 17.BisecttheangelRST.
33
Top Related