Unit 2 Transformations / Rigid Motions Lesson 1: Reflections on … · 2018-08-08 · 1 Unit 2...

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Unit2Transformations/RigidMotions

Lesson1:ReflectionsontheCoordinatePlaneOpeningExerciseWhatdoyourememberaboutreflections???Takethepoint(4,2)andreflectitasstated.Plotthenewpointandstateitscoordinates.Reflectioninthex-axis

CoordinatesoftheNewPoint:_____________

Reflectioninthey-axisCoordinatesoftheNewPoint:_____________Reflectionintheliney=x

CoordinatesoftheNewPoint:_____________

Reflectioninthelinex=1

CoordinatesoftheNewPoint:_____________

SummaryoftheRules:

rx−axis : (x, y) →ry−axis : (x, y) →ry= x : (x, y) →

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VocabularyAtransformationisachangeintheposition,shape,orsizeofafigure.Arigidmotionisatransformationthatchangesonlythepositionofthefigure(lengthandanglemeasuresarepreserved).Animageistheresultofatransformationofafigure(calledthepre-image).Toidentifytheimageofapoint,useprimenotation.TheimageofpointAisA’(readasAprime).Example1GivenΔABC withverticesA(-5,1),B(-1,1)andC(-1,7).a. GraphΔABC ontheaxesprovidedbelow.b. Onthesamesetofaxes,graphΔA 'B 'C ' ,theimageofΔABC reflectedoverthe

x-axis.c. Onthesamesetofaxes,graphΔA ''B ''C '' ,theimageofΔABC reflectedoverthe

y-axis.

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Example2GivenΔABC withverticesA(2,3),B(0,6)andC(2,6).a. GraphΔABC ontheaxesprovidedbelow.b. GraphandstatethecoordinatesofΔA 'B 'C ' ,theimageofΔABC reflectedoverthe

liney=x.c. GraphandstatethecoordinatesofΔA ''B ''C '' ,theimageofΔA 'B 'C ' reflectedover

theline y = −2 .

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Example3ΔDOG hasverticesD(-1,1),O(-2,5)andG(-5,2)andΔD 'O 'G ' hasverticesD’(3,-3),O’(7,-4)andG’(4,-7).a. Graphandlabel andΔD 'O 'G ' b. Graphtheline y = x − 2 c. Whatistherelationshipbetweenthelineofreflectionandthesegmentsconnecting

thecorrespondingpoints?(ThinkbacktoUnit1)

ΔDOG

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Homework1. Usingthepoint(-5,3),finditsimageafterthefollowingreflections(theuseofthe

gridisoptional).

a. rx−axis b. ry−axis c. ry= x d. rx=−1 e. ry=2

2. GivenΔABC withverticesA(2,1),B(3,4)andC(-4,5).

a. GraphΔABC ontheaxesprovided.

b. GraphandstatethecoordinatesofΔA 'B 'C ' ,theimageofΔABC reflectedoverthex-axis.

c. GraphandstatethecoordinatesofΔA ''B ''C '' ,theimageofΔA 'B 'C ' reflectedovertheline y = x .

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Lesson2:ReflectionsofftheCoordinatePlaneOpeningExerciseYouwillneedacompassandastraightedgeAsshowninthediagramtotheright,ΔABC isreflectedacrossDEandmapsontoΔA 'B 'C ' .a. Useyourstraightedgetodrawin segments AA ' , BB ' andCC ' .b. Useyourcompasstomeasurethe

distancesfromthepre-imagepointtoDEandfromtheimagepointtoDE.Whatdoyounoticeaboutthesedistances?

c. WhatistherelationshipbetweensegmentDEandeachofthesegmentsthatwere

drawninparta?

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Example1YouwillneedacompassandastraightedgeWenowknowthatthelineofreflectionistheperpendicularbisectorofthesegmentsconnectingthepre-imagetotheimagepoint.Wearegoingtousethis,alongwithourknowledgeofconstructions,toconstructthelineofreflection.a. Connectanypointtoitsimagepoint.b. Drawtheperpendicularbisectorofthis segment.Thisisthelineofreflection!Eachpointanditsimagepointareequidistantfromthisline!!!Selectingasecondpairofpointsandconstructingitsperpendicularbisectorcanverifythis.

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ExercisesYouwillneedacompassandastraightedgeConstructthelineofreflectionforeachimageanditspre-image.1.2.

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Example2 YouwillneedacompassandastraightedgeUsingourknowledgeofperpendicularbisectorswearegoingtoreflectanobjectoveragivenline.ReflectΔABC overDE.Canwethinkofanotherwaytodothissameproblem?

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HomeworkYouwillneedacompassandastraightedgeIn1-2,constructthelineofreflection.1. 2.

3. Reflectthegivenfigureacrossthelineofreflectionprovided.

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Lesson3:TranslationsontheCoordinatePlaneOpeningExerciseAnothertypeofrigidmotioniscalledatranslation,orslide,inwhicheverypointofafigureismovedthesamedistanceinthesamedirection.DescribehowtotranslateΔABC toitsimageΔA 'B 'C ' .Wecanalsowritethistranslationusingtwodifferenttypesofnotation:Isthesizeoftheobjectpreservedunderatranslation?Istheorderoftheverticesthesame?VocabularyAnisometryisatransformationthatdoesnotchangeinsize.Theseincludealloftherigidmotions:reflections,translationandrotations.Adirectisometrypreservessizeandtheorder(orientation)ofthevertices.Anoppositeisometrypreservesthesize,buttheorderoftheverticeschanges.Atranslationwouldbeanexampleofwhichisometry?Areflectionwouldbeanexampleofwhichisometry?

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Example1UsingthesametranslationastheOpeningExercise,wearegoingtoexplorethepaththeimagefollows.Usingrays,connectthepre-imagepointswiththeimagepoints.Whatdoyounoticeabouttheraysyouhavedrawn?Thisrayiscalledavector.Avectorisadirectedlinesegmentthathasbothlengthanddirection.ExercisesYouwillneedastraightedgeDrawthevectorthatdefineseachtranslationbelow.

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TranslationRule:Ta,b (x, y)→ Exercises1. Determinethecoordinatesoftheimageofthepoint(5,-3)underT-2,-1.

2. Determinethecoordinatesoftheimageofthepoint(-8,-3)underthetranslation

(x, y)→ (x + 4, y −1) 3. Determinethetranslationthatmapsthepoint(-5,5)tothepoint(7,1).4. Atranslationmapsthepoint(-2,5)tothepoint(-4,-4).Whatistheimageof(1,4)

underthesametranslation?5. Translatetheimageoneunitdownandthreeunitsright.Drawthevectorthat

definesthetranslation.

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Homework1. Determinethecoordinatesoftheimageofthepoint(-2,2)underT-2,6.

2. Determinethecoordinatesoftheimageofthepoint(2,-3)underthetranslation

(x, y)→ (x − 4, y + 2) 3. IftranslationTx,y mapspointP(-3,1)ontopointP’(5,5),findxandy.4. Atranslationmapsthepoint(3,1)tothepoint(-4,2).Whatistheimageof(4,-1)

underthesametranslation?5. ΔABC hasverticesA(1,1),B(2,3)andC(6,-2).

a. GraphΔABC b. GraphΔA 'B 'C ' ,theimageof

ΔABC afterthetranslation (x, y)→ (x − 2, y − 6)

c. Drawthevectorthatdefinesthe translation.

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Lesson4:TranslationsofftheCoordinatePlaneOpeningExerciseYouwillneedacompassandastraightedgeConstructΔA 'B 'C ' ,theimageofΔABC afterareflectionoverlinel.

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Example1YouwillneedacompassandastraightedgeAswelearnedinourlastlesson,atranslatedobjectfollowsthepathofavector.Ifweweretoconnecteachpre-imagepointwithitsimagepoint,wewouldhavecongruentandparallelsegments.Inthediagrambelow,segmentABistranslatedtoproduceA’B’.a. Drawthevectorthatdefinesthistranslation.b. Usingyourcompass,locateB’c. ConstructsegmentA’B’

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Example2YouwillneedacompassandastraightedgeApply TAB

u ruu tosegmentCD .Vectorstellus2things:distance&directionStep1:LengthUsingyourcompass,measurehowfar willtravelbasedonvector AB

! "!!.

Step2:DirectionHowwillyouslidepointCtoitsnewpoint?

CD

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Example3YouwillneedacompassandastraightedgeApply TAB

u ruu toΔXYZ .Example4Useyourcompassandstraightedgetoapply tothecirclebelow:

TABu ruu

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R S

TU

Homework1. a. Translatethefigure2unitsdownand3units

left

b. Drawthevectorthatdefinesthetranslation2. UseyourcompassapplyTMNu ruuu topentagonABCDE.

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Lesson5:RotationsontheCoordinatePlaneOpeningExerciseFillinthetablebelowtoidentifythecharacteristicsandtypesofrigidmotionsbeingappliedtocreateΔA 'B 'C ' ,theimageofΔABC .

TypeofRigidMotion:

Issizepreserved?

Isorientationpreserved?

Whichtypeofisometry?

VocabularyArotationisarigidmotionthatturnsafigureaboutafixedpointcalledthecenterofrotation.Theangleofrotationisthenumberofdegreesthefigurerotates.Apositiveangleofrotationturnsthefigurecounterclockwise(anegativeangleofrotationcanbeusedforclockwiserotations).

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Example1Takethepoint(4,2)androtateitasstated.Plotthenewpointandstateitscoordinates.Rotationof90 °

CoordinatesoftheNewPoint:_____________

Rotationof180 °

CoordinatesoftheNewPoint:_____________Rotationof270 °

CoordinatesoftheNewPoint:_____________

Rotationof360 °

CoordinatesoftheNewPoint:_____________

SummaryoftheRules:

RO, 90° : (x, y) →RO, 180° : (x, y) →RO, 270° : (x, y) →

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Example2Given withverticesC(-1,2),O(-1,5)andW(-3,3).a. Graph ontheaxesprovided.b. Graphandstatethecoordinatesof

,theimageof afterarotationof

c. Whattypeofisometryistheimage?

ΔCOW

ΔCOW

ΔC 'O 'W ' ΔCOW180°

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Example3Wearenowgoingtorotateanimagearoundapointotherthantheorigin.GivenΔABC withverticesA(2,3),B(0,6)andC(2,6).a. GraphΔABC ontheaxesprovidedbelow.b. GraphandstatethecoordinatesofΔA 'B 'C ' ,theimageofΔABC afterarotationof

90° aboutthepoint(-1,2).Torotateanimageaboutapointotherthantheorigin:Step1:Translatetherotationpointtotheorigin.Step2:Translatethepre-imageusingthesametranslationasStep1.Step3:Rotatetheimagefollowingtherulesofrotations.Step4:TranslatetheimagetheoppositedirectionasthetranslationfromStep1.

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Exercises1. GivenΔABC withverticesA(4,0),B(2,3)andC(1,2).

a. GraphΔABC ontheaxesprovided.

b. Graphandstatethecoordinatesof ΔA 'B 'C ' ,theimageofΔABC after

ry−axis .

c. Graphandstatethecoordinatesof ΔA"B"C" ,theimageofΔA 'B 'C ' after rx−axis .

d. Thistwice-reflectedobjectisthe sameaswhichsingle transformation? 2. Given withverticesD(-3,5),O(4,6)andG(0,2).

a. Graph ontheaxesprovided.

b. Graphandstatethecoordinatesof,theimageof after

arotationof aboutthepoint(3,-2).

ΔDOG

ΔDOG

ΔD 'O 'G ' ΔDOG90°

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Homework1. GivenΔCAT withverticesC(-1,5),A(-3,1)andT(-2,-2).

a. GraphΔCAT ontheaxesprovided.

b. Graphandstatethecoordinatesof ΔC 'A 'T ' ,theimageofΔCAT after

RO,270° .

c. Graphandstatethecoordinatesof ΔC"A"T " ,theimageofΔC 'A 'T ' after rx−axis .

d. Whichtypeofisometryistheimage inpartc?Explainyouranswer.

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Lesson6:RotationsofftheCoordinatePlaneOpeningExerciseRotatethefigurebelowaboutpointA.Showtheimageofthefigureafterrotationsof90° ,180° and270° .NowrotatethefigureaboutpointDusingthesamerotationsof90° ,180° and270° .Noticethedifferencebetweenthetwoimagescreated!!!

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Example1Giventhetwofiguresbelow,wearegoingtouseaprotractortomeasuretheangleofrotation.Tofindtheangleofrotation:Step1:Identifythepointthatisthecenterofrotation.Step2:Measuretheangleformedbyconnectingcorrespondingverticestothecenterpoint

ofrotation.Step3:Checkyouranswerusingadifferentsetofcorrespondingvertices.

CenterofRotation: CenterofRotation:AngleofRotation: AngleofRotation:

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Example2Wearenowgoingtolocatethecenterofrotation.Step1:ConstructtheperpendicularbisectorofsegmentAA’.Step2:ConstructtheperpendicularbisectorofsegmentBB’.Step3:Thepointofintersectionoftheperpendicularbisectorsisthepointofrotation.

LabelthispointP.

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Example3InUnit1,welookedattheconstructionofahexagonbyusingequilateraltriangles.Usingthissameconcept,wearegoingtorotateΔABC 60° aroundpointFusingacompassandstraightedgeonly.Howcouldwerotatethisimage120° ?Howcouldwerotatethisimage90° ?

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Homework1. Findthecenterofrotationandtheangleofrotationforthetransformationbelow:2. Rotate 120° aroundpointRusingacompassandstraightedgeonly.

ΔABC

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Lesson7:TypesofSymmetryOpeningExerciseΔA 'B 'C ' istheimageofΔABC afterareflectionacrossline l1 .a. Reflecttheimageacrossline l2 .b. Whatistherelationshipbetweentheoriginaltriangleandthetwice-reflected

image?c. WhatdoespointRrepresent?d. Howcouldwedeterminetheangleofrotation?Reflectingafiguretwiceoverintersectinglineswillgivethesameresultasarotationaboutthepointofintersection!

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Example1LookingattheOpeningExercise,wecanseethatthelinesofreflectionarealsolinesofsymmetry.Thelineofsymmetryisequidistantfromallcorrespondingpairsofpoints.Inthefiguresbelow,sketchallthelinesofsymmetry:

Example2RotationalSymmetryisarotationthatmapsafigurebackontoitself.Inregularpolygons(polygonsinwhichallsidesandanglesarecongruent)thenumberofrotationalsymmetriesisequaltothenumberofsidesofthefigure.Howcanwefindtheanglesofrotation? Equilateral

Triangle Square RegularPentagon

RegularHexagon

#ofsides

AnglesofRotation

Arotationof 360° willalwaysmapafigurebackontoitself.Thisiscalledtheidentitytransformation.

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Example3Whenreflectinganobjectthroughapoint,theimageandthepre-imagecreatepointsymmetry.Withpointsymmetry,theobjectwilllookexactlythesameupsidedown!Thiscanbeseenbyreflectinganobjectthroughtheoriginaspicturedtotheright.Whichoftheobjectsbelowhavepointsymmetry?

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ExercisesUsingregularpentagonABCDEpicturedtotheright,completethefollowing:1. Drawalllineofsymmetry.2. Locatethecenterofrotationalsymmetry.3. Describeallsymmetriesexplicitly.

a. Whatkindsarethere?

b. Howmanyarerotations?

c. Whataretheanglesofrotation?d. Howmanyarereflections?

AB

C

D

E

AB

C

D

E S

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Homework1. Usingthefiguretotheright,describeallthesymmetriesexplicitly.

a. Howmanyarerotations?

b. Whataretheanglesofrotation?

c. Howmanyarereflections?

d. Shadethefiguresothattheresultingfigureonlyhas3possiblerotationalsymmetries.

2. Usingthefiguresprovided,shadeexactly2ofthe9smallersquaressothatthe

resultingfigurehas: Onlyoneverticalandone Onlytwolinesofsymmetry

horizontallineofsymmetry aboutthediagonals Onlyonehorizontalline Nolineofsymmetry ofsymmetry

A B

CD

A B

CD

S

A B

CD

A B

CD

S

A B

CD

A B

CD

S

A B

CD

A B

CD

S

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Lesson8:CompositionsofRigidMotionsOpeningExerciseLookingatthepictureabove,describethetypeofrigidmotionthattakesplacetogofrom: Δ1→ Δ2 : Δ2→ Δ3 : Δ3→ Δ4 :Whenaseriesofrigidmotionstakesplacewithonerigidmotionbuildingoffanother(asshownabove)thisiscalledacomposition.Thesymbolusedforcompositions:Whenperformingorwritingacomposition,youmustworkfromrighttoleft!Twodifferentwaystowritethecompositionpicturedabove: 1. 2.

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Example1PentagonABCDEispicturedtotherightwherelandmarelinesofsymmetry.Evaluatethefollowingcompositions: a. rl ! rm (E)

b. rm ! rl BA( ) c. rm ! R72° (B) Example2ThecoordinatesofΔABC areA(-2,2),B(3,5)andC(4,2).GraphandstatethecoordinatesofΔA ''B ''C '' ,theimageofΔABC afterthecomposition ry−axis !T2,−3 .Acompositionofatranslationandareflectioniscalledaglidereflection.

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Example3Usingthediagrams,writetheruleofthecomposition:a. b. c. d. e. Ineachofthecompositionsshowninpartsa-d,istheimagecongruenttothepre-

image?Explain.

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A B

CD

E

G H

J I

A

B C

X

Y

Z

A B

CD

A B

CD

BA

CD

E

A B

CD

Homework1. ThecoordinatesofΔABC areA(-1,1),B(-5,3)andC(-2,7).Graphandstatethe

coordinatesofΔA ''B ''C '' ,theimageofΔABC afterthecompositionT−5, 1 !RO, 180° .2. Inthediagrampictured,allofthesmallertrianglesarecongruenttoeachother.

Whatrigidmotion(s)mapZBontoAZ?3. Inthediagrampictured,whatrigidmotion(s)mapCDontoAB?

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A B

CD

G H

IJ

Lesson9:CongruenceandCorrespondenceOpeningExercisePicturedbelowaresquareABCDandrhombusGHIJ.Aretheycongruent?Explain.VocabularyWhenfiguresarecongruent,thismeansthatthereisarigidmotion(oracompositionofrigidmotions)thatmapsthepre-imageontotheimage.Thisrigidmotioniscalledacongruence.Example1Underthisdefinitionofcongruence,describewhythefiguresintheOpeningExercisearenotcongruent.

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VocabularyAcorrespondencebetweentwotrianglesisapairingofeachvertexofonetrianglewithoneandonlyonevertexofanothertriangle.Thispairingcanbeexpandedtofiguresotherthantrianglesandcouldalsoinvolvesides.Example2Inthefigurebelow,thetriangleonthelefthasbeenmappedtotheoneontherightbyarotationof 240° aboutP.Identifyallsixpairsofcorrespondingparts(anglesandsides).

a. IsΔABC ≅ ΔXYZ ?Explain.b. WhatrigidmotionmappedΔABC ontoΔXYZ ?Writethetransformationin

functionnotation.ImportantDiscovery!Rigidmotionsproducecongruentfiguresandtherefore,congruentparts(anglesandsides).Asaresult,wecansaythatcorrespondingpartsofcongruentfiguresarecongruent.

Correspondingangles Correspondingsides

∠A→ AB→

∠B→ AC→

∠C→ BC→

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A B

CD

G H

IJ

A B

CD

A

B C

X

Y Z

Exercises1. ABCDisasquare,andACisonediagonalofthesquare.ΔABC isa

reflectionofΔADC acrosssegmentAC.

a. Completethetablebelowidentifyingthecorrespondinganglesandsides.

Correspondingangles Correspondingsides

∠BAC→∠ABC→∠BCA→

AB→BC→AC→

b. Arethecorrespondingsidesandanglescongruent?Justifyyourresponse.c. IsΔABC ≅ ΔADC ?Justifyyourresponse.

2. EachsideofΔXYZ istwicethelengthofeachsideofΔABC .

a. Fillintheblanksbelowsothateachrelationshipbetweenlengthsofsidesis true.

__________× 2 =__________

__________× 2 =__________

__________× 2 =__________

b. IsΔABC ≅ ΔXYZ ?Justifyyourresponse. ImportantDiscovery!Correspondingpartsdonotalwaysresultincongruentfigures.

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Example3ThefiguretotherightshowsaseriesofrigidmotionsperformedonΔABC toproducethedottedtriangle.a. Identifytherigidmotions.b. LabelvertexA’’.c. IsΔA ''B ''C '' ≅ ΔABC ?Justifyyouranswer.Example4CompletethetablebasedontheseriesofrigidmotionsperformedonΔABC toproducethedottedtriangle.

Sequenceofrigidmotions

Compositioninfunctionnotation

Sequenceofcorrespondingsides

Sequenceofcorrespondingangles

Trianglecongruencestatement

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Homework1. Usingyourunderstandingofcongruence,explainthefollowing:

a. Whyisatrianglenotcongruenttoaquadrilateral?

b. Whyisanisoscelestrianglenotcongruenttoascalenetriangle?2. Drawadiagramwithtwotrianglesinwhichallthreecorrespondinganglesare

congruentbutthecorrespondingsidesarenotcongruent.3. Inthefigurebelow,thetriangleonthelefthasbeenmappedtotheoneontheright

byarotationof80° aboutvertexC.Identifyallsixpairsofcorrespondingparts(anglesandsides).

Writetherigidmotioninfunctionnotation.

Correspondingangles Correspondingsides

∠A→ AB→

∠B→ AC→

∠C→ BC→