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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.2
Mathematics Vision Project
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mathematicsvisionproject.org
6.2
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READY Topic:BasicanglerelationshipsMatchthediagramsbelowwiththebestnameorphrasethatdescribestheangles.______1. _______2.
______3. ______4.
_____5. ______6.
a. AlternateInteriorAngles b. VerticalAngles c. ComplementaryAngles
d. TriangleSumTheorem e. LinearPair f. SameSideInteriorAngles
READY, SET, GO! Name Period Date
Page 10
A B
C F
E D
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. 3 Similar Triangles and OtherFigures
A Solidify Understanding Task
Twofiguresaresaidtobecongruentifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations.InMathematicsIwefoundthatweonlyneededthreepiecesofinformationtoguaranteethattwotriangleswerecongruent:SSS,ASAorSAS.
WhataboutAAA?Aretwotrianglescongruentifallthreepairsofcorrespondinganglesarecongruent?Inthistaskwewillconsiderwhatistrueaboutsuchtriangles.
Part1DefinitionofSimilarity:Twofiguresaresimilarifthesecondcanbeobtainedfromthefirst
byasequenceofrotations,reflections,translations,anddilations.
MasonandMiaaretestingoutconjecturesaboutsimilarpolygons.Hereisalistoftheirconjectures.
Conjecture1:Allrectanglesaresimilar.
Conjecture2:Allequilateraltrianglesaresimilar.
Conjecture3:Allisoscelestrianglesaresimilar.
Conjecture4:Allrhombusesaresimilar.
Conjecture5:Allsquaresaresimilar.
1. Whichoftheseconjecturesdoyouthinkaretrue?Why?
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Not all 23442 15882TrueNottrue
NottrueTrue
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MasonisexplainingtoMiawhyhethinksconjecture1istrueusingthediagramgivenbelow.
“Allrectangleshavefourrightangles.IcantranslateandrotaterectangleABCDuntilvertexAcoincideswithvertexQinrectangleQRST.Since∠Aand∠Qarebothrightangles,sideABwilllieontopofsideQR,andsideADwilllieontopofsideQT.IcanthendilaterectangleABCDwithpointAasthecenterofdilation,untilpointsB,C,andDcoincidewithpointsR,S,andT.
2. DoesMason’sexplanationconvinceyouthatrectangleABCDissimilartorectangleQRSTbasedonthedefinitionofsimilaritygivenabove?Doeshisexplanationconvinceyouthatallrectanglesaresimilar?Whyorwhynot?
MiaisexplainingtoMasonwhyshethinksconjecture2istrueusingthediagramgivenbelow.
“Allequilateraltriangleshavethree60°angles.IcantranslateandrotateΔABCuntilvertexAcoincideswithvertexQinΔQRS.Since∠Aand∠Qareboth60°angles,sideABwilllieontopofsideQR,andsideACwilllieontopofsideQS.IcanthendilateΔABCwithpointAasthecenterofdilation,untilpointsBandCcoincidewithpointsRandS.”
3. DoesMia’sexplanationconvinceyouthatΔABCissimilartoΔQRSbasedonthedefinitionofsimilaritygivenabove?Doesherexplanationconvinceyouthatallequilateraltrianglesaresimilar?Whyorwhynot?
4. Foreachoftheotherthreeconjectures,writeanargumentlikeMason’sandMia’stoconvincesomeonethattheconjectureistrue,orexplainwhyyouthinkitisnotalwaystrue.
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DBD c
From B to Rmaynotbesameasscalefactoras D to T
Thedilationmatching B to R will be sameasfrom CtoS sincesides of equilateral D'sare
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
a. Conjecture3:Allisoscelestrianglesaresimilar.
b. Conjecture4:Allrhombusesaresimilar.
c. Conjecture5:Allsquaresaresimilar.
Whilethedefinitionofsimilaritygivenatthebeginningofthetaskworksforallsimilarfigures,analternativedefinitionofsimilaritycanbegivenforpolygons:Twopolygonsaresimilarifallcorrespondinganglesarecongruentandallcorrespondingpairsofsidesareproportional.
5. HowdoesthisdefinitionhelpyoufindtheerrorinMason’sthinkingaboutconjecture1?
6. HowdoesthisdefinitionhelpconfirmMia’sthinkingaboutconjecture2?
7. Howmightthisdefinitionhelpyouthinkabouttheotherthreeconjectures?
a. Conjecture3:Allisoscelestrianglesaresimilar.
b. Conjecture4:Allrhombusesaresimilar.
c. Conjecture5:Allsquaresaresimilar.
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sincescalefactorsforthedilationthatmatchesBtoRandDtotmaynotbeequaltheratioABTate isnotnecessarilyequaltoAyattherefore correspondingsidesneednotbeproportional
sincescalefactorsforthedilationthatmatchesBtoRandCtoS are equaltheratioABTate is equalto AYastherefore correspondingsidesneedtobeproportional
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Part2(AAA,SASandSSSSimilarity)
Fromourworkabovewithrectanglesitisobviousthatknowingthatallrectangleshavefourrightangles(anexampleofAAAAforquadrilaterals)isnotenoughtoclaimthatallrectanglesaresimilar.Whatabouttriangles?Ingeneral,aretwotrianglessimilarifallthreepairsofcorrespondinganglesarecongruent?
8. Decideifyouthinkthefollowingconjectureistrue.
Conjecture:Twotrianglesaresimilariftheircorrespondinganglesarecongruent.
9. Explainwhyyouthinktheconjecture—twotrianglesaresimilariftheircorrespondinganglesarecongruent—istrue.Usethefollowingdiagramtosupportyourreasoning,Remembertostartbymarkingwhatyouaregiventobetrue(AAA)inthediagram.
Hint:BeginbytranslatingAtoD.
10. Miathinksthefollowingconjectureistrue.Shecallsit“AASimilarityforTriangles.”Whatdoyouthink?Isittrue?Why?
Conjecture:Twotrianglesaresimilariftheyhavetwopairofcorrespondingcongruentangles.
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True
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
11. Usingthediagramgiveninproblem9,howmightyoumodifyyourproofthatΔABC~ΔDEFifyouaregiventhefollowinginformationaboutthetwotriangles:
a. ∠A≅∠D,!" = ! ⋅ !", !" = ! ⋅ !" ;thatis, !"!" = !"!"
b. !" = ! ⋅ !", !" = ! ⋅ !" and !" = ! ⋅ !";thatis, !"!" =!"!" =
!"!"
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SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6.3
Needhelp?Visitwww.rsgsupport.org
READY Topic:SolvingproportionsinmultiplewaysSolveeachproportion.Showyourworkandcheckyoursolution.
1. 2. 3.34 =
!20
!7 =
1821
36 =
8!
4. 5. 6.9! =
610
34 =
! + 320
712 =
!24
7. 8. 9.!2 =
1320
3! + 2 =
65
32 = 12
!
SETTopic:ProvingShapesaresimilarProvideanargumenttoproveeachconjecture,orprovideacounterexampletodisproveit.
10. Allrighttrianglesaresimilar 11. Allregularpolygonsaresimilartootherregularpolygonswiththesamenumberofsides.
12. Thepolygonsonthegridbelowaresimilar. 13. Thepolygonsonthegridbelowaresimilar.
READY, SET, GO! Name Period Date
Page 20
64044115 X
ft 67222bt
SECONDARY MATH II // MODULE 6
SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6.3
Needhelp?Visitwww.rsgsupport.org
Asequenceoftransformationsoccurredtocreatethetwosimilarpolygons.Provideaspecificsetofstepsthatcanbeusedtocreatetheimagefromthepre-image.14. 15.
16. 17.
GO Topic:RatiosinsimilarpolygonsForeachpairofsimilarpolygonsgivethreeratiosthatwouldbeequivalent.18. 19.
20. 21.
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