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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
7. 4 Congruent Triangles
A Solidify Understanding Task
Weknowthattwotrianglesarecongruentifallpairsofcorrespondingsidesarecongruent
andallpairsofcorrespondinganglesarecongruent.Wemaywonderifknowinglessinformation
aboutthetriangleswouldstillguaranteetheyarecongruent.
Forexample,wemaywonderifknowingthattwoanglesandtheincludedsideofone
trianglearecongruenttothecorrespondingtwoanglesandtheincludedsideofanothertriangle—a
setofcriteriawewillrefertoasASA—isenoughtoknowthatthetwotrianglesarecongruent.And,
ifwethinkthisisenoughinformation,howmightwejustifythatthiswouldbeso.
HereisadiagramillustratingASAcriteriafortriangles:
1. Basedonthediagram,whichanglesarecongruent?Whichsides?
2. Toconvinceourselvesthatthesetwotrianglesarecongruent,whatelsewouldweneedto
know?
3. Usetracingpapertofindasequenceoftransformationsthatwillshowwhetherornotthese
twotrianglesarecongruent.
4. Listyoursequenceoftransformationshere:
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
Yoursequenceoftransformationsisenoughtoshowthatthesetwotrianglesarecongruent,
buthowcanweguaranteethatallpairsoftrianglesthatshareASAcriteriaarecongruent?
Perhapsyoursequenceoftransformationslookedlikethis:
• translatepointAuntilitcoincideswithpointR• rotate aboutpointRuntilitcoincideswith • reflectΔABCacross
Nowthequestionis,howdoweknowthatpointChastoland
onpointTafterthereflection,makingallofthesidesandanglescoincide?
5. AnswerthisquestionasbestyoucantojustifywhyASAcriteriaguaranteestwotrianglesarecongruent.Toanswerthisquestion,itmaybehelpfultothinkabouthowyouknowpointCcan’tlandanywhereelseintheplaneexceptontopofT.
Usingtracingpaper,experimentwiththeseadditionalpairsoftriangles.Trytodetermineif
youcanfindasequenceoftransformationsthatwillshowifthetrianglesarecongruent.Becareful,
theremaybesomethataren’t.Ifthetrianglesappeartobecongruentbasedonyour
experimentation,writeanargumenttoexplainhowyouknowthatthistypeofcriteriawillalways
work.Thatis,whatguaranteesthattheunmarkedsidesoranglesmustalsocoincide?
6. Givencriteria:__________ Arethetrianglescongruent?_________
Listyourtransformations Ifthetrianglesarecongruent,justifywhythisintheorderperformed: willalwaysbetruebasedonthiscriteria:
�
AB
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RS
�
RS
Wecanusetheword“coincides”whenwewanttosaythattwopointsorlinesegmentsoccupythesamepositionontheplane.Whenmakingargumentsusingtransformationswewillusethewordalot.
21
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
7. Giveninformation:__________ Arethetrianglescongruent?_________
Listyourtransformations Ifthetrianglesarecongruent,justifywhythisintheorderperformed: willalwaysbetruebasedonthiscriteria:
8. Giveninformation:__________ Arethetrianglescongruent?_________
Listyourtransformations Ifthetrianglesarecongruent,justifywhythisintheorderperformed: willalwaysbetruebasedonthiscriteria:
22
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
9. Giveninformation:__________ Arethetrianglescongruent?_________
Listyourtransformations Ifthetrianglesarecongruent,justifywhythisintheorderperformed: willalwaysbetruebasedonthiscriteria:
10. Basedontheseexperimentsandyourjustifications,whatcriteriaorconditionsseemtoguaranteethattwotriangleswillbecongruent?Listasmanycasesasyoucan.MakesureyouincludeASAfromthetrianglesweworkedwithfirst.
11. YourfriendwantstoaddAAStoyourlist,eventhoughyouhaven’texperimentedwiththis
particularcase.Whatdoyouthink?ShouldAASbeaddedornot?Whatconvincesyouthatyouarecorrect?
12. YourfriendalsowantstoaddHL(hypotenuse-leg)toyourlist,eventhoughyouhaven’t
experimentedwithrighttrianglesatall,andyouknowthatSSAdoesn’tworkingeneralfromproblem8.Whatdoyouthink?ShouldHLforrighttrianglesbeaddedornot?Whatconvincesyouthatyouarecorrect?
23
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
7.4
READY Topic:Correspondingpartsoffiguresandtransformations.Giventhefiguresineachsketchwithcongruentanglesandsidesmarked,firstlistthepartsofthefiguresthatcorrespond(Forexample,in#1,∠ ! ≅ ∠!)Thendetermineifareflectionoccurredaspartofthesequenceoftransformationsthatwasusedtocreatetheimage.
1.
Congruencies
∠ ! ≅ ∠ !
Reflected?YesorNo
2.
Congruencies
Reflected?YesorNo
READY, SET, GO! Name PeriodDate
T
SR
C
A
B
Z
W
X
Y
F
EH
G
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SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
7.4
SET Topic:TriangleCongruenceExplainwhetherornotthetrianglesarecongruent,similar,orneitherbasedonthemarkingsthatindicatecongruence.
3.
4.
5. 6.
7.
8.
Usethegivencongruencestatementtodrawandlabeltwotrianglesthathavethepropercorrespondingpartscongruenttooneanother.
9.∆ !"# ≅ ∆ !"# 10.∆ !"# ≅ ∆ !"#
25
SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.4
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
7.4
GO Topic:Solvingequationsandfindingrecursiverulesforsequences.Solveeachequationfort.11.!!!!! = 5 12.10 − ! = 4! + 12 − 3!13.! = 5! − ! 14.!" − ! = 13! + !Usethegivensequenceofnumbertowritearecursiveruleforthenthvalueofthesequence.15.5,15,45,… 16. 1
2, 0, - 1
2, -1, ...
17.3,-6,12,-24,… 18. 1
2, 1
4, 1
8, ...
26