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IntegratedMath1HonorsModule6Honors

Transformations,Congruence,andConstructions

Adaptedfrom

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutorius

©2012MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense

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SDUHSDMath1Honors

Module6HonorsOverview

PrerequisiteConcepts&Skills: ApplyPythagoreanTheorem Graphlinearandexponentialfunctions Identify/solveforslopeandx‐andy‐interceptsoflinearfunctions Solvemulti‐stepequations Identifybasicgeometricshapesandcharacteristics Solvingsystemsofequations

SummaryoftheConcepts&SkillsinModule6H:

Developdefinitionsofrigid‐motiontransformations:translations,rotations,andreflections Examineslopeofperpendicularandparallellines Examinewhichrigidmotiontransformationcarryoneimageontoanothercongruentimage Writeandapplyformaldefinitionsoftherigid‐motiontransformations Findrotationalsymmetryandlinesofsymmetryinquadrilaterals Examinecharacteristicsofregularpolygonsthatemergefromrotationalsymmetryandlinesof

symmetry Makeandjustifypropertiesofquadrilateralsusingsymmetrytransformations Describeasequenceoftransformationsthatwillcarrycongruentimagesontoeachother EstablishtheASA,SAS,andSSScriteriaforcongruenttriangles Explorecompassandstraightedgeconstructions Writeproceduresforcompassandstraightedgeconstructionsandwhyitcreatesthedesired

object(s)ContentStandardsandStandardsofMathematicalPracticeCovered:

ContentStandards:G.CO.1,G.CO.2,G.CO.3,G.CO.4,G.CO.5,G.CO.6,G.CO.7,G.CO.8,G.CO.12,G.CO.13,G.GPE.5

StandardsofMathematicalPractice:1. Makesenseofproblems&persevereinsolvingthem2. Attendtoprecision3. Reasonabstractly&quantitatively4. Constructviablearguments&critiquethereasoningofothers5. Modelwithmathematics6. Useappropriatetoolsstrategically7. Lookfor&makeuseofstructure8. Lookfor&expressregularityinrepeatedreasoning

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Module6HVocabulary: PythagoreanTheorem Construction Proof Quadrilateral Rhombus Equilateral Parallelogram Square Trapezoid Polygon Diagonal Rotation Reflection Transformation Translation Lineofsymmetry Lineofreflection Rotationalsymmetry Triangle Pentagon Hexagon Heptagon Octagon Congruent Similar Inscribed

ConceptsUsedintheNextModule:

Usecoordinatestofinddistancesanddeterminetheperimeterofgeometricshapes Proveslopecriteriaforparallelandperpendicularlines Usecoordinatestoalgebraicallyprovegeometrictheorems Writetheequation bycomparingparallellinesandfindingk Determinethetransformationfromonefunctiontoanother Translatelinearandexponentialfunctionsusingmultiplerepresentations Thearithmeticofvectorsandsolvingproblemsinvolvingquantitiesthatcanberepresentedby

vectors Matrices–propertiesofaddition,multiplication,identityandinverseproperties,findingthe

determinant,andsolvingasystemusingthemultiplicativeinversematrix Usingmatrixmultiplicationtoreflectandrotatevectorsandimages

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Module6Honors–Transformations,Congruence,andConstructionsNote:Module6HonorshasbeendividedintotwopartssothatPart2canbestartedsecondsemester/termifdesired.Iftimeisavailableduringfirstsemester/term,itisrecommendedtocontinuethroughandfinishtheentiremoduleduringfirstsemester/term.Part1includestransformationsandcongruence,whilePart2bringsinconstructions.Part1:TransformationsandCongruence6.1HDevelopingthedefinitionsoftherigid‐motiontransformations:translations,reflectionsandrotationsandExaminingtheslopeofperpendicularlines(G.CO.1,G.CO.4,G.CO.5)WarmUp:LeapingLizards!‐ADevelopUnderstandingTaskClassroomTask:IsItRight?‐ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.1H 6.2HDeterminingwhichrigid‐motiontransformationscarryoneimageontoanothercongruentimageandWritingandapplyingformaldefinitionsoftherigid‐motiontransformations:translations,reflectionsandrotations(G.CO.1,G.CO.2,G.CO.4,G.GPE.5)WarmUp:LeapFrog–ASolidifyUnderstandingTaskClassroomTask:LeapYear–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.2H6.3HFindingrotationalsymmetryandlinesofsymmetryinspecialtypesofquadrilateralsandExaminingcharacteristicsofregularpolygonsthatemergefromrotationalsymmetryandlinesofsymmetry(G.CO.3,G.CO.6)WarmUp:SymmetriesofQuadrilaterals–ADevelopUnderstandingTaskClassroomTask:SymmetriesofRegularPolygons–ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.3H6.4HMakingandjustifyingpropertiesofquadrilateralsusingsymmetrytransformations(G.CO.3,G.CO.4,G.CO.6)ClassroomTask:Quadrilaterals‐BeyondDefinition–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.4H6.5HDescribingasequenceoftransformationsthatwillcarrycongruentimagesontoeachotherandEstablishingtheASA,SASandSSScriteriaforcongruenttriangles(G.CO.5,G.CO.6,G.CO.7,G.CO.8)WarmUp:Sharesolutionsto6.4HReadySetGoquestion#28ClassroomTask:CongruentTriangles–ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.5H6.6HUsingASA,SAS,orSSStodetermineiftwotrianglesembeddedinanothergeometricfigurearecongruent.(G.CO.7,G.CO.8)WarmUp:DefiningbisectorsofanglesandperpendicularbisectorsClassroomTask:CongruentTrianglestotheRescue–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.6H

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Part2–Constructions6.7HExploringcompassandstraightedgeconstructionstoconstructrhombusesandsquares(G.CO.12,G.CO.13)ClassroomTask:UnderConstruction–ADevelopUnderstandingTaskReady,Set,GoHomework:Constructions6.7H6.8HExploringcompassandstraightedgeconstructionstoconstructparallelograms,equilateraltrianglesandinscribedhexagonsandExploringcompassandstraightedgeconstructionstoconstructparallelograms,equilateraltrianglesandinscribedhexagons(G.CO.12,G.CO.13)Warm‐Up:GeometricconstructionsusingcompassandstraightedgeandconstructingtransformationsClassroomTask:ConstructionBasics–ASolidifyUnderstandingTaskReady,Set,GoHomework:Constructions6.8H6.9HWritingproceduresforcompassandstraightedgeconstructions(G.CO.12,G.CO.13)WarmUp:ConstructionBlueprints–APracticeUnderstandingTaskModule6ReviewClassroomTask:CarouselActivityModule6ReviewHomeworkIntrotoModule7HonorsReady,Set,GoModule6HonorsChallengeProblems

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©2012http://www.clker.com

/clipart‐green‐gecko

6.1HLeapingLizards!ADevelopUnderstandingTaskAnimatedfilmsandcartoonsarenowusuallyproducedusingcomputertechnology,ratherthanthehand‐drawnimagesofthepast.Computeranimationrequiresbothartistictalentandmathematicalknowledge.Sometimesanimatorswanttomoveanimagearoundthecomputerscreenwithoutdistortingthesizeandshapeoftheimageinanyway.Thisisdoneusinggeometrictransformationssuchastranslations(slides),reflections(flips),androtations(turns)orperhapssomecombinationofthese.Thesetransformationsneedtobepreciselydefined,sothereisnodoubtaboutwherethefinalimagewillenduponthescreen.Sowheredoyouthinkthelizard,shownonthegridsonthefollowingpages,willendupusingthefollowingtransformations?(Theoriginallizardwascreatedbyplottingthefollowinganchorpointsonthecoordinategridandthenlettingacomputerprogramdrawthelizard.Theanchorpointsarealwayslistedinthisorder:tipofnose,centerofleftfrontfoot,belly,centerofleftrearfoot,pointoftail,centerofrearrightfoot,back,centeroffrontrightfoot.)Originallizardanchorpoints:12, 12 , 15, 12 , 17, 12 , 19, 10 , 19, 14 , 20, 13 , 17, 15 , 14, 16

Eachstatementbelowdescribesatransformationoftheoriginallizard.Foreachofthestatements:

Plottheanchorpointsforthelizardinitsnewlocation. Connectthepre‐imageandimageanchorpointswithlinesegments,orcirculararcs,whicheverbest

illustratestherelationshipbetweenthem.LazyLizardTranslatetheoriginallizardsothepointatthetipofitsnoseislocatedat 24, 20 ,makingthelizardappeartobesunbathingontherock.LungingLizardRotatethelizard90°(counterclockwise)aboutpointA 12, 7 soitlookslikethelizardisdivingintothepuddleofmud.LeapingLizardReflectthelizardaboutgivenline 16soitlookslikethelizardisdoingabackflipoverthecactus.

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LazyLizard(Translation)

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LungingLizard(Rotation)

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LeapingLizard(Reflection)

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2012 www.flickr.com/photos/juggernau

tco/  

6.1HIsItRight?ASolidifyUnderstandingTaskInLeapingLizards,youprobablythoughtalotaboutperpendicularlines,particularlywhenrotatingthelizardabouta90°angleorreflectingthelizardacrossaline.Inprevioustasks,wehavemadetheobservationthatparallellineshavethesameslope.Inthistask,wewillmakeobservationsabouttheslopesofperpendicularlines.PerhapsinLeapingLizardsyouusedaprotractororsomeothertoolorstrategytohelpyoumakearightangle.Inthistaskweconsiderhowtocreatearightanglebyattendingtoslopesonthecoordinategrid.Webeginbystatingafundamentalideaforourwork:Horizontalandverticallinesareperpendicular.Forexample,onacoordinategrid,thehorizontalline 2andtheverticalline

3intersecttoformfourrightangles.

Butwhatifalineorlinesegmentisnothorizontalorvertical?Howdowedeterminetheslopeofaline,orlinesegment,thatwillbeperpendiculartoit?Experiment11. Considerthepoints 2, 3 and 4, 7 andthelinesegment, ,betweenthem.Whatistheslopeofthislinesegment?

2. Locateathirdpoint , onthecoordinategrid,sothepoints 2, 3 , 4, 7 and , formtheverticesofarighttriangle,with asitshypotenuse.

3. Explainhowyouknowthatthetriangleyouformedcontainsarightangle?

4. Nowrotatethisrighttriangle90°counterclockwise

aboutthevertexpoint 2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.

5. Comparetheslopeofthehypotenuseofthisrotatedrighttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?

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Experiment2Repeatsteps1‐5fromexperiment1forthepoints 2, 3 and 5, 4 .

1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformedcontains

arightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint

2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.

5. Comparetheslopeofthehypotenuseofthisrotatedrighttrianglewiththeslopeofthehypotenuseofthe

pre‐image.Whatdoyounotice?

Experiment3Repeatsteps1‐5forthepoints 2, 3 and 7, 5 .1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformedcontains

arightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint

2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.

5. Comparetheslopeofthehypotenuseofthisrotated

righttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?

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Experiment4Repeatsteps1‐5forthepoints 2, 3 and 0, 6 .1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformed

containsarightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint

2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.

5. Comparetheslopeofthehypotenuseofthisrotated

righttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?

Basedonexperiments1‐4,stateanobservationabouttheslopesofperpendicularlines.Whilethisobservationisbasedonafewspecificexamples,canyoucreateanargumentorjustificationforwhythisisalwaystrue?(Note:Youwillexamineaformalproofofthisobservationinthenextmodule.)

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2012http://openclipart.org/detail/33781/architetto 

6.2HWarmUp:LeapFrogASolidifyUnderstandingTaskJoshisanimatingascenewhereatroupeoffrogsisauditioningfortheAnimalChannelrealityshow,"TheBayou'sGotTalent".Inthisscene,thefrogsaredemonstratingtheir"leapfrog"acrobaticsact.Joshhascompletedafewkeyimagesinthissegment,andnowneedstodescribethetransformationsthatconnectvariousimagesinthescene.Foreachpre‐image/imagecombinationlistedbelow,describethetransformationthatmovesthepre‐imagetothefinalimage.

Ifyoudecidethetransformationisarotation,youwillneedtogivethecenterofrotation,thedirectionoftherotation(clockwiseorcounterclockwise),andthemeasureoftheangleofrotation.

Ifyoudecidethetransformationisareflection,youwillneedtogivetheequationofthelineof

reflection.

Ifyoudecidethetransformationisatranslation,youwillneedtodescribethe"rise"and"run"betweenpre‐imagepointsandtheircorrespondingimagepointsorwriteatranslationrule.

Ifyoudecideittakesacombinationoftransformationstogetfromthepre‐imagetothefinalimage,

describeeachtransformationintheordertheywouldbecompleted.

Pre‐image FinalImage Description

1. image1 image2

2. image2 image3

3. image3 image4

4. image1 image5

5. image2 image4

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2012  w

ww.flickr.com/photos/suendercafe 

6.2HLeapYearAPracticeUnderstandingTaskCarlosandClaritaarediscussingtheirlatestbusinessventurewiththeirfriendJuanita.Theyhavecreatedadailyplannerthatisbotheducationalandentertaining.Theplannerconsistsofapadof365pagesboundtogether,onepageforeachdayoftheyear.Theplannerisentertainingsinceimagesalongthebottomofthepagesformaflip‐bookanimationwhenthumbedthroughrapidly.Theplanneriseducationalsinceeachpagecontainssomeinterestingfacts.Eachmonthhasadifferenttheme,andthefactsforthemonthhavebeenwrittentofitthetheme.Forexample,thethemeforJanuaryisastronomy,thethemeforFebruaryismathematics,andthethemeforMarchisancientcivilizations.CarlosandClaritahavelearnedalotfromresearchingthefactstheyhaveincluded,andtheyhaveenjoyedcreatingtheflip‐bookanimation.ThetwinsareexcitedtosharetheprototypeoftheirplannerwithJuanitabeforesendingittoprinting.Juanita,however,hasamajorconcern."Nextyearisleapyear,"sheexplains,"youneed366pages."SonowCarlosandClaritahavethedilemmaofhavingtocreateanextrapagetoinsertbetweenFebruary28andMarch1.Herearetheplannerpagestheyhavealreadydesigned.

February28Acircleisthesetofallpointsinaplanethatareequidistantfromafixedpointcalledthecenterofthecircle.Anangleistheunionoftworaysthatshareacommonendpoint.Anangleofrotationisformedwhenarayisrotatedaboutitsendpoint.Theraythatmarksthepre‐imageoftherotationisreferredtoasthe“initialray”andtheraythatmarkstheimageoftherotationisreferredtoasthe“terminalray.”Angleofrotationcanalsorefertothenumberofdegreesafigurehasbeenrotatedaboutafixedpoint,withacounterclockwiserotationbeingconsideredapositivedirectionofrotation.

March1Whyarethere360°inacircle?Onetheoryisthatancientastronomersestablishedthatayearwasapproximately360days,sothesunwouldadvanceinitspathrelativetotheearlyapproximately ofaturn,oronedegree,eachday.(The5extradaysinayearwereconsideredunluckydays.)AnothertheoryisthattheBabyloniansfirstdividedacircleintopartsbyinscribingahexagonconsistingof6equilateraltrianglesinsideacircle.Theanglesoftheequilateraltrianglelocatedthecenterofthecirclewerefurtherdividedinto60equalparts,sincetheBabyloniannumbersystemwasbase‐60(insteadofbase‐10likeournumbersystem).Anotherreasonfor360°inacirclemaybethefactthat360has24divisors,soacirclecaneasilybedividedintomanysmaller,equal‐sizedparts.

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Part1SincethethemeforthefactsforFebruaryismathematics,Claritasuggeststhattheywriteformaldefinitionsofthethreerigid‐motiontransformationstheyhavebeenusingtocreatetheimagesfortheflip‐bookanimation.Howwouldyoucompleteeachofthefollowingdefinitions?Usethefollowingwordsandphrasesinyourdefinitions:perpendicularbisector,centerofrotation,equidistant,angleofrotation,concentriccircles,parallel,image,pre‐image,preservesdistanceandanglemeasures.1. Atranslationofasetofpointsinaplane...2. Arotationofasetofpointsinaplane...3. Areflectionofasetofpointsinaplane...4. Translations,rotationsandreflectionsarerigidmotiontransformationsbecause...

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Part2InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimageinthemiddleoftheirflip‐bookanimation.TheanimationsequenceisofDorothy'shousespinningfromtheWizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.CarlosbelievesthathecangetfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,andthenreflectingitagain.Usingtheresourcepage,verifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28andMarch1worksasheintended.Forexample,5. WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegiven

lineofreflection?6. WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenline

ofreflection?

7. WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?

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6.2HLeapYearResourcePage

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2012 www.flickr.com/photos/temaki/ 

6.3HWarmUp:SymmetriesofQuadrilateralsADevelopUnderstandingTaskAlinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedontoitselfbyarotationissaidtohaverotationalsymmetry.Everyfour‐sidedpolygonisaquadrilateral.Somequadrilateralshaveadditionalpropertiesandaregivenspecialnameslikesquares,parallelograms,andrhombuses.Adiagonalofaquadrilateralisformedwhenoppositeverticesareconnectedbyalinesegment.Inthistask,youwilluserigid‐motiontransformationstoexplorelinesymmetryandrotationalsymmetryinvarioustypesofquadrilaterals.1. Aparallelogramisaquadrilateralinwhichbothpairsofoppositesidesareparallel.Isitpossibleto

reflectorrotateaparallelogramontoitself?

Fortheparallelogramshownatright,find

anylinesofreflection,or anycentersandanglesofrotation

Describetherotationsand/orreflectionsthatcarryaparallelogramontoitself.Beasspecificaspossibleinyourdescriptions.

2. Arectangleisaparallelogramthatcontainsfourrightangles.Isitpossibletoreflectorrotatea

rectangleontoitself?

Fortherectangleshownatright,find

anylinesofreflection,or anycentersandanglesofrotation

Describetherotationsand/orreflectionsthatcarryarectangleontoitself.Beasspecificaspossibleinyourdescriptions.

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3. Arhombusisaparallelograminwhichallsidesarecongruent.Isitpossibletoreflectorrotatearhombusontoitself?

Fortherhombusshownatright,find

anylinesofreflection,or anycentersandanglesofrotation

Describetherotationsand/orreflectionsthatcarryarhombusontoitself.Beasspecificaspossibleinyourdescriptions.

4. Asquareisaparallelogramwithallsidescongruentandallanglescongruent.Isitpossibletoreflector

rotateasquareontoitself?

Forthesquareshownatright,find

anylinesofreflection,or anycentersandanglesofrotation

Describetherotationsand/orreflectionsthatcarryasquareontoitself.Beasspecificaspossibleinyourdescriptions.

5. Atrapezoidisaquadrilateralwithonlyonepairofoppositesidesparallel.Isitpossibletoreflector

rotateatrapezoidontoitself?

Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind

anylinesofsymmetry,or anycentersofrotationalsymmetry

Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.

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2012 www.flickr.com/photos/tamburix  

6.3HSymmetriesofRegularPolygonsASolidifyUnderstandingTaskAlinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedontoitselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanylinesegmentthatconnectsnon‐consecutiveverticesofthepolygon.Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitontoitself.Beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation.1. Anequilateraltriangle

Rotations Reflections

2. Asquare

Rotations Reflections

3. Aregularpentagon

Rotations Reflections

   

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4. Aregularhexagon

Rotations Reflections

5. Aregularoctagon

Rotations Reflections

6. Aregularnonagon

Rotations Reflections

7. Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina

regularpolygon?8. Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational

symmetryinaregularpolygon?

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6.4HQuadrilaterals—BeyondDefinitionAPracticeUnderstandingTaskWehavefoundthatmanydifferentquadrilateralspossesslineand/orrotationalsymmetry.1. Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin

termsoftheirsymmetries.

2. Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand

highlightedinthestructureoftheabovechart?

2012 www.flickr.com/photos/gabby‐girl

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Basedonthesymmetrieswehaveobservedinvarioustypesofquadrilaterals,wecanmakeclaimsaboutotherfeaturesandpropertiesthatthequadrilateralsmaypossess.3. Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutparallelogramsbesidesthedefiningpropertythatoppositesidesofaparallelogramareparallel?Makealistofadditionalpropertiesofparallelogramsthatseemtobetruebasedonthetransformation(s)oftheparallelogramontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.4. Arectangleisaparallelogramthatcontainsfourrightangles.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutrectanglesbesidesthedefiningpropertythatallfouranglesarerightangles?Makealistofadditionalpropertiesofrectanglesthatseemtobetruebasedonthetransformation(s)oftherectangleontoitself.Youwillwanttoconsiderpropertiesofthesides,theangles,andthediagonals.

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5. Arhombusisaparallelograminwhichallfoursidesarecongruent.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutarhombusbesidesthedefiningpropertythatallsidesarecongruent?Makealistofadditionalpropertiesofrhombusesthatseemtobetruebasedonthetransformation(s)oftherhombusontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.

6. Asquareisaparallelogramwithallsidescongruentandallanglescongruent.

Basedonwhatyouknowabouttransformations,whatcanwesayaboutasquare?Makealistofpropertiesofsquaresthatseemtobetruebasedonthetransformation(s)ofthesquaresontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.

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7. Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsoftheirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeofquadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.

8. Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristicsand

highlightedinthestructureoftheabovechart?9. Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?

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2012 www.flickr.com/photos/ shaireproductions  6.5HCongruentTriangles

ASolidifyUnderstandingTaskPartI:DrawSpecificTriangles–Foreachcategorylistedbelow,drawatrianglethatmatchesthedescription.Eachtriangleshouldbedrawnonaseparatepieceofpattypaper.Labeleachcharacteristiconthepattypaper.

A. Threesides–Drawasegment4cmlong.Usethecompasstomaketointersectingarcsof5cmand6cmfromoppositeendpoints.Connectpointofintersectiontoendpointsofthesegmenttoformatriangle.

B. Threeangles–Drawanangleof35°andextendtherays(thesewillbecomesidesofthetriangle).Drawa65°angleononeoftheraysandextendtointersecttheotherrayofthe35°angle.Whatisthemeasureofthethirdangle?Labelthisonthetriangle.

C. Twosidesandanincludedangle–Drawatrianglesuchthattwosideshavelengthsof4cmand7cmandtheanglebetweenis70°.

D. Twosidesandanon‐includedangle–Drawatrianglesuchthattwosideshavelengths6cmand7cmandtheangleNOTbetweenis55°.

E. Twoanglesandanincludedside–Drawatrianglesuchthattwoangleshavemeasures75°and40°andthesidebetweenthemhaslength5cm.

Rotategroups(1movesup,1movesdownsothatnooriginalpartnersaretogether).Partnerwithsomeonefromadifferentgroup.

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PartIICompareTrianglesfromtheSameCategory–Determinewhichcategories(AthroughE)producetrianglesthatarecongruent.Recalculateanymeasurementsofnon‐congruenttrianglestoverify.Listthedescriptionsofthetypesofcongruenttriangles:

PartIIIProvingCongruenceThroughRigid‐MotionTransformations–EachpersonchoosesonepairofcongruenttriangleslistedinPartII.

A. FoldthegraphpaperinhalftomakeQ1andQ2.Drawinthex‐axisalongthebottomandthey‐axisalongthefold.

B. TransferbothimagesfromthepattypaperontothegraphpaperbyplacingonecornerofthepattypaperattheorigininQ1andtheotherpattypaperattheorigininQ2.

C. Labelandconfirmthemeasurementsusingarulerandprotractor.D. LabeltheverticesofoneofthetrianglesasA,B,andC.E. Writeasequenceoftransformationstocarry∆ABContotheothertriangle.Usecoloredpencilsto

showeachindividualtransformationinthesequence.F. Exchangepaperstoverifythatyourpartner’ssequenceoftransformationsaccuratelydemonstrate

thecongruenceofthetriangles.

PartIVReflection–ReflectonwhatyoudidinPartIII.Howweretransformationsusedtoverifycongruence?Willthisholdtrueforalltriangleswithinthesamecategory?Explain.

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6.6HWarmUpDefiningbisectorsofanglesandperpendicularbisectors

1. Basedonthemeaningof“bisect”,whichmeanstosplitintotwoequalparts,whatwoulditmeantobisect

anangle?Describeinwordsandalsoprovidevisualstocommunicatethemeaningofanglebisector.

2. Whatdoesitmeanifyouhaveaperpendicularbisectorofalinesegment?Providebothwritten

explanationandvisualsketchestocommunicatethemeaningofperpendicularbisector.

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2012 www.flickr.com/ photos/aren

amontanus 6.6HCongruentTrianglestotheRescue

APracticeUnderstandingTaskPart1ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent.Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertexpointoftheanglemadeupofthetwocongruentsides,andthemidpointofthethirdside.Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineofsymmetry?Ifso,

whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabouthisstatement?2. WhatelsewouldZacneedtoknowaboutthelinethroughthevertexpointoftheanglemadeupofthe

twocongruentsidesandthemidpointofthethirdsideinordertoknowthatitisalineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)

3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)createstwo

congruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASorSSS—couldsheusetosupportthisclaim?Describethesidesand/oranglesyouthinkarecongruent,andexplainhowyouknowtheyarecongruent.

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4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoesthatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenotformedbythetwocongruentsides)?

5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoesthatimply

aboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthisline—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non‐congruentsideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthevertexangleformedbythetwocongruentsides.)

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Part2LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotationalsymmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinition,youmadesomeobservationsaboutsides,anglesanddiagonalsofvarioustypesofquadrilateralsbasedonyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbefurtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZacandSionedidintheirworkwithisoscelestriangles.Pickoneofthefollowingquadrilateralstoexplore:

Arectangleisaparallelogramthatcontainsfourrightangles. Arhombusisaparallelograminwhichallsidesarecongruent. Asquareisaparallelogramwithfourrightanglesandallsidesarecongruent

1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthequadrilateral

A,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.2. Basedon(a)yourdrawing,(b)thegivendefinitionofyourquadrilateral,and(c)informationaboutsides

andanglesthatyoucangatherbasedonlinesofreflectionandrotationalsymmetry,listasmanypairsofcongruenttrianglesasyoucanfindinthetableonthenextpage.

Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused(ASA,SASorSSS)todeterminethatthetwotrianglesarecongruent,andexplainhowyouknowthattheanglesand/orsidesrequiredbythecriteriaarecongruent.

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3. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethecongruent

trianglestojustifysomethingelseaboutthequadrilateral,suchas:

thediagonalsarecongruent thediagonalsareperpendiculartoeachother thediagonalsbisecttheanglesofthequadrilateral

PickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandwriteanargumentthatwouldconvinceZacandSionethatthestatementistrue.

CongruentTrianglesCriteriaUsed(ASA,SAS,SSS)

Reasonsthesidesand/oranglesarecongruent.

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6.7HUnderConstructionADevelopUnderstandingTaskInancienttimes,oneoftheonlytoolsbuildersandsurveyorshadforlayingoutaplotoflandorthefoundationofabuildingwasapieceofrope.Therearetwogeometricfiguresyoucancreatewithapieceofrope:youcanpullittighttocreatealinesegment,oryoucanfixoneend,andwhileextendingtheropetoitsfulllengthtraceoutacirclewiththeotherend.Geometricconstructionshavetraditionallymimickedthesetwoprocessesusinganunmarkedstraightedgetocreatealinesegmentandacompasstotraceoutacircle(orsometimesaportionofacirclecalledanarc).Usingonlythesetwotoolsyoucanconstructavarietyofgeometricshapes.Supposeyouwanttoconstructarhombususingonlyacompassandstraightedge.Youmightbeginbydrawingalinesegmenttodefinethelengthofaside,anddrawinganotherrayfromoneoftheendpointsofthelinesegmenttodefineanangle,asinthefollowingsketch.

Nowthehardworkbegins.Wecan’tjustkeepdrawinglinesegments,becausewehavetobesurethatallfoursidesoftherhombusarethesamelength.Thisiswhenourconstructiontoolscomeinhandy.ConstructingarhombusKnowingwhatyouknowaboutcirclesandlinesegments,howmightyoulocatepointContherayinthediagramabovesothedistancefromBtoCisthesameasthedistancefromBtoA?1. DescribehowyouwilllocatepointCandhowyouknow ≅ ,thenconstructpointConthediagram

above.Nowthatwehavethreeofthefourverticesoftherhombus,weneedtolocatepointD,thefourthvertex.2. DescribehowyouwilllocatepointDandhowyouknow ≅ ≅ ,thenconstructpointDonthe

diagramabove.

2012 www.flickr.com/ photos/subflux  

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ConstructingaSquare(Arhombuswithrightangles)Theonlydifferencebetweenconstructingarhombusandconstructingasquareisthatasquarecontainsrightangles.Therefore,weneedawaytoconstructperpendicularlinesusingonlyacompassandstraightedge.Wewillbeginbyinventingawayto“construct”aperpendicularbisectorofalinesegment.3. Given below,foldandcreasethepapersothatpointRisreflectedontopointS.Basedonthe

definitionofreflection,whatdoyouknowaboutthis“creaseline”?

Youhave“constructed”aperpendicularbisectorof byusingapaper‐foldingstrategy.Isthereawaytoconstructthislineusingacompassandstraightedge?4. Experimentwiththecompasstoseeifyoucandevelopastrategytolocatepointsonthe“creaseline”.

Whenyouhavelocatedatleasttwopointsonthe“creaseline”usethestraightedgetofinishyourconstructionoftheperpendicularbisector.Describeyourstrategyforlocatingpointsontheperpendicularbisectorof .

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Nowthatyouhavecreatedalineperpendicularto wewillusetherightangleformedtoconstructasquare.5. Labelthemidpointof onthediagramaboveaspointM.Usingsegment asonesideofthesquare,

andtherightangleformedbysegment andtheperpendicularlinedrawnthroughpointMasthebeginningofasquare.Finishconstructingthissquareonthediagramabove.(Hint:Rememberthatasquareisalsoarhombus,andyouhavealreadyconstructedarhombusinthefirstpartofthistask.)

6. Likearhombus,anequilateraltrianglehasthreecongruentsides.Showanddescribehowyouwould

locatethethirdvertexpointonanequilateraltriangle,given belowasonesideoftheequilateraltriangle.

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ConstructingaParallelogram7. Toconstructaparallelogramwewillneedtobeabletoconstructalineparalleltoagivenlinethrougha

givenpoint.Forexample,supposewewanttoconstructalineparalleltosegment throughpointConthediagrambelow.Sincewehaveobservedthatparallellineshavethesameslope,alinethroughpointCwillbeparallelto onlyiftheangleformedby andthelineweconstructiscongruentto∠ .CanyoudescribeandillustrateastrategythatwillconstructananglewithvertexatpointCandasideparallelto ?(Hint:Weknowthatcorrespondingpartsofcongruenttrianglesarecongruent,soperhapswecanbeginbyconstructingsomecongruenttriangles.)

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ConstructingaHexagonInscribedinaCircleBecauseregularpolygonshaverotationalsymmetry,theycanbeinscribedinacircle.Thecircumscribedcirclehasitscenteratthecenterofrotationandpassesthroughalloftheverticesoftheregularpolygon.Wemightbeginconstructingahexagonbynoticingthatahexagoncanbedecomposedintosixcongruentequilateraltriangles,formedbythreeofitslinesofsymmetry.8. Sketchadiagramofsuchadecomposition.9. Basedonyoursketch,whereisthecenterofthecirclethatwouldcircumscribethehexagon?10.Thesixverticesofthehexagonlieonthecircleinwhichtheregularhexagonisinscribed.Thesixsidesof

thehexagonarechordsofthecircle.Howarethelengthsofthesechordsrelatedtothelengthsoftheradiifromthecenterofthecircletotheverticesofthehexagon?Beabletojustifyhowyouknowthisisso.

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11.Basedonthisanalysisoftheregularhexagonanditscircumscribedcircle,constructanddescribeyourprocessforahexagoninscribedinthecirclegivenbelow.

12.Modifyyourworkwiththehexagontoconstructanequilateraltriangleinscribedinthecirclegiven

below.

13.Describehowyoumightconstructasquareinscribedinacircle.

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6.8HWarmUpGeometricconstructionsusingcompassandstraightedgeandconstructingtransformations1. Constructaparallelogramgivensides and

and∠ .

2. Constructalineparallelto throughpointR.

Ineachproblembelowusecompassandstraightedgetoconstructthetransformationthatisdescribed.3. Construct∆ ′ ′ ′sothatitisatranslationof∆ .(Hint:parallellinesmaybeuseful.)

4. Construct∆ ′ ′ ′sothatitisareflectionof∆ overlinem.(Hint:perpendicularlinesmaybeuseful.)

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© Elenathew

ise #4322186 

6.8HConstructionBasicsASolidifyUnderstandingTask1. UsingyourcompassdrawseveralconcentriccirclesthathavepointAasacenterandthen

drawthosesamesizedconcentriccirclesthathaveBasacenter.WhatdoyounoticeaboutwhereallthecircleswithcenterAintersectallthecorrespondingcircleswithcenterB?

2. Intheproblemaboveyouhavedemonstratedonewaytofindthemidpointofalinesegment.Explain

anotherwaythatalinesegmentcanbebisectedwithouttheuseofcircles.3. Bisecttheanglebelowfirstwithpaperfolding,thenwithcompassandstraightedge.

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4. Copythesegmentbelowusingconstructiontoolsofcompassandstraightedge,labeltheimageD’E’.

5. Copytheanglebelowusingconstructiontoolofcompassandstraightedge.

6. Constructarhombuswithside thatisnotasquare.Besuretocheckthatyourfinalfigureisa

rhombus.

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7. Constructasquarewithsidelength .Besuretocheckthatyourfinalfigureisasquare.

8. Givensegment showallpointsCsuchthat∆ isanisoscelestriangle,with asthebase

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9. Givensegment showallpointsCsuchthat∆ isarighttriangle.

10. Giventheequilateraltrianglebelow,findthecenterofrotationofthetriangleusingcompassand

straightedge.

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6.10HConstructionBlueprintsAPracticeUnderstandingTaskForeachofthefollowingstraightedgeandcompassconstructions,illustrateorlistthestepsforcompletingtheconstructionandgiveanexplanationforwhytheconstructionworks.Yourexplanationsmaybebasedonrigid‐motiontransformations,congruenttriangles,orpropertiesofquadrilaterals.

Purposeoftheconstruction PerformtheconstructionIllustrationand/orstepsforcompletingtheconstruction

Justificationofwhythisconstructionworks

Copyingasegment

1. Set the span of the compass tomatchthedistancebetweenthetwoendpointsofthesegment.

2. Withoutchangingthespanofthecompass,drawanarconaraycenteredattheendpointoftheray.Thesecondendpointofthesegmentiswherethearcintersectstheray.

Thegivensegmentandtheconstructedsegmentareradiiofcongruentcircles.

Copyinganangle

   

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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction

Justificationofwhythisconstructionworks

Bisectingasegment

Bisectinganangle

Constructingaperpendicularbisectorofalinesegment

   

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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction

Justificationofwhythisconstructionworks

Constructingaperpendiculartoalinethroughagivenpoint

Constructingalineparalleltoagivenlinethroughagivenpoint

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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction

Justificationofwhythisconstructionworks

Constructinganequilateraltriangle

Constructingaregularhexagoninscribedinacircle

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EndofModule6HonorsChallengeProblemsThefollowingproblemsareintendedforstudentstoworkonafterModule6HTest.Theproblemsfocusonusingsimilartrianglestofindarea.ThenextmodulebuildsontheideaofconnectingAlgebraandGeometry.Thefollowingpageisblankfortheteachertocopyandgivetoeachstudentafterthetest.Belowarethesolutions.BothrighttriangleABCandisoscelestriangleBCD,shownhere,haveheight5cmfrombase 12cm.Usethefigureandinformationprovidedtoanswerthefollowingquestions.

1. WhatistheabsolutedifferencebetweentheareasofΔABCandΔBCD?2. WhatistheratiooftheareaofΔABEtoΔCDE?3. WhatistheareaofΔBCE?4. WhatistheareaofpentagonABCDE?