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→