SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.4

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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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BY

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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

�

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

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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