Module 7 - Mathematics Vision Project

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 7

Congruence, Construction & Proof

SECONDARY

MATH ONE

An Integrated Approach

SECONDARY MATH 1 // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

MODULE 7 - TABLE OF CONTENTS

CONGRUENCE, CONSTRUCTION AND PROOF

7.1 Under Construction – A Develop Understanding Task

Exploring compass and straightedge constructions to construct rhombuses and squares

(G.CO.12, G.CO.13)

READY, SET, GO Homework: Congruence, Construction and Proof 7.1

7.2 More Things Under Construction – A Develop Understanding Task

Exploring compass and straightedge constructions to construct parallelograms, equilateral triangles and

inscribed hexagons (G.CO.12, G.CO.13)


7.3 Can You Get There From Here? – A Develop Understanding Task

Describing a sequence of transformations that will carry congruent images onto each other (G.CO.5)


7.4 Congruent Triangles – A Solidify Understanding Task

Establishing the ASA, SAS and SSS criteria for congruent triangles (G.CO.6, G.CO.7, G.CO.8)


7.5 Congruent Triangles to the Rescue – A Practice Understanding Task

Identifying congruent triangles and using them to justify claims (G.CO.7, G.CO.8)


7.6 Justifying Constructions – A Solidify Understanding Task

Examining why compass and straightedge constructions produce the desired results (G.CO.12, G.CO.13)


SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.1




7. 1 Under Construction

A Develop Understanding Task

Anciently,oneoftheonlytoolsbuildersandsurveyorshadforlayingoutaplotoflandorthefoundationofabuildingwasapieceofrope.

Therearetwogeometricfiguresyoucancreatewithapieceofrope:youcanpullittighttocreatealinesegment,oryoucanfixoneend,and—whileextendingtheropetoitsfulllength—traceoutacirclewiththeotherend.Geometricconstructionshavetraditionallymimickedthesetwoprocessesusinganunmarkedstraightedgetocreatealinesegmentandacompasstotraceoutacircle(orsometimesaportionofacirclecalledanarc).Usingonlythesetwotoolsyoucanconstructallkindsofgeometricshapes.

Supposeyouwanttoconstructarhombususingonlyacompassandstraightedge.Youmightbeginbydrawingalinesegmenttodefinethelengthofaside,anddrawinganotherrayfromoneoftheendpointsofthelinesegmenttodefineanangle,asinthefollowingsketch.

Nowthehardworkbegins.Wecan’tjustkeepdrawinglinesegments,becausewehavetobesurethatallfoursidesoftherhombusarethesamelength.Wehavetostopdrawingandstartconstructing.

CC

BY

UK

Dep

artm

ent

for

Inte

rnat

iona

l Dev

elop

men

t

http

s://f

lic.k

r/p/

ekU

mnG

1






ConstructingarhombusKnowingwhatyouknowaboutcirclesandlinesegments,howmightyoulocatepointContherayinthediagramabovesothedistancefromBtoCisthesameasthedistancefromBtoA?

1. DescribehowyouwilllocatepointCandhowyouknow ,thenconstructpointConthediagramabove.

Nowthatwehavethreeofthefourverticesoftherhombus,weneedtolocatepointD,thefourthvertex.

2. DescribehowyouwilllocatepointDandhowyouknow ,thenconstructpointDonthediagramabove.

ConstructingaSquare(Arhombuswithrightangles)

Theonlydifferencebetweenconstructingarhombusandconstructingasquareisthatasquarecontainsrightangles.Therefore,weneedawaytoconstructperpendicularlinesusingonlyacompassandstraightedge.

Wewillbeginbyinventingawaytoconstructaperpendicularbisectorofalinesegment.

3. Given!"below,foldandcreasethepapersothatpointRisreflectedontopointS.Basedonthedefinitionofreflection,whatdoyouknowaboutthis“creaseline”?

�

BC ≅ BA

�

CD ≅ DA ≅ AB

2






Youhave“constructed”aperpendicularbisectorof!"byusingapaper-foldingstrategy.Is

thereawaytoconstructthislineusingacompassandstraightedge?

4. Experimentwiththecompasstoseeifyoucandevelopastrategytolocatepointsonthe“creaseline”.Whenyouhavelocatedatleasttwopointsonthe“creaseline”usethestraightedgetofinishyourconstructionoftheperpendicularbisector.Describeyourstrategyforlocatingpointsontheperpendicularbisectorof!".

Nowthatyouhavecreatedalineperpendicularto!" wewillusetherightangleformedto

constructasquare.

5. Labelthemidpointof!" onthediagramaboveaspointM.Usingsegment!"asonesideofthesquare,andtherightangleformedbysegment!"andtheperpendicularlinedrawnthroughpointMasthebeginningofasquare.Finishconstructingthissquareonthediagramabove.(Hint:Rememberthatasquareisalsoarhombus,andyouhavealreadyconstructedarhombusinthefirstpartofthistask.)

3






7.1

READY Topic:Toolsforconstructionandgeometricwork.

1.UsingyourcompassdrawseveralconcentriccirclesthathavepointAasacenterandthendrawthosesamesizedconcentriccirclesthathaveBasacenter.WhatdoyounoticeaboutwhereallthecircleswithcenterAintersectallthecorrespondingcircleswithcenterB?

2.Intheproblemaboveyouhavedemonstratedonewaytofindthemidpointofalinesegment.Explainanotherwaythatalinesegmentcanbebisectedwithouttheuseofcircles.

SET Topic:Constructionswithcompassandstraightedge.3.Bisecttheanglebelowdoitwithcompassandstraightedgeaswellaswithpaperfolding.

READY, SET, GO! Name PeriodDate

4






7.1

4.Copythesegmentbelowusingconstructiontoolsofcompassandstraightedge,labeltheimageD’E’.5.Copytheanglebelowusingconstructiontoolofcompassandstraightedge.

6.ConstructarhombusonthesegmentABthatisgivenbelowandthathaspointAasavertex.Besuretocheckthatyourfinalfigureisarhombus.

5






7.1

7.ConstructasquareonthesegmentCDthatisgivenbelow.Besuretocheckthatyourfinalfigureisasquare.8.Giventheequilateraltrianglebelow,findthecenterofrotationofthetriangleusingcompassandstraightedge.

GO Topic:SolvingsystemsofequationsSolveeachsystemofequations.Utilizesubstitution,elimination,graphingormatrices.

9. ! = 11 + !2! + ! = 19

10. −4! + 9! = 9! − 3! = −6

11. ! + 2! = 11! − 4! = 2

12. ! = −! + 1! = 2! + 1

13. ! = −2! + 7−3! + ! = −8

14. 4! − ! = 7−6! + 2! = 8

6






7. 2 More Things Under

Construction


Likearhombus,anequilateraltrianglehasthreecongruentsides.Showanddescribehow

youmightlocatethethirdvertexpointonanequilateraltriangle,given!"belowasonesideoftheequilateraltriangle.

ConstructingaParallelogram

Toconstructaparallelogramwewillneedtobeabletoconstructalineparalleltoagiven

linethroughagivenpoint.Forexample,supposewewanttoconstructalineparalleltosegment

!"throughpointConthediagrambelow.Sincewehaveobservedthatparallellineshavethesameslope,thelinethroughpointCwillbeparallelto!"onlyiftheangleformedbythelineand!"iscongruentto∠ABC.Canyoudescribeandillustrateastrategythatwillconstructananglewith

vertexatpointCandasideparallelto!"?

CC

BY

Bri

an N

egus

http

s://f

lic.k

r/p/

7eEb

DP

7






ConstructingaHexagonInscribedinaCircle

Becauseregularpolygonshaverotationalsymmetry,theycanbeinscribedinacircle.The

circumscribedcirclehasitscenteratthecenterofrotationandpassesthroughalloftheverticesof

theregularpolygon.

Wemightbeginconstructingahexagonbynoticingthatahexagoncanbedecomposedinto

sixcongruentequilateraltriangles,formedbythreeofitslinesofsymmetry.

1. Sketchadiagramofsuchadecomposition.

2. Basedonyoursketch,whereisthecenterofthecirclethatwouldcircumscribethehexagon?

3. Thesixverticesofthehexagonlieonthecircleinwhichtheregularhexagonisinscribed.Thesixsidesofthehexagonarechordsofthecircle.Howarethelengthsofthesechordsrelatedtothelengthsoftheradiifromthecenterofthecircletotheverticesofthehexagon?Thatis,howdoyouknowthatthesixtrianglesformedbydrawingthethreelinesofsymmetryareequilateraltriangles?(Hint:Consideringanglesofrotation,canyouconvinceyourselfthatthesesixtrianglesareequiangular,andthereforeequilateral?)

8






4. Basedonthisanalysisoftheregularhexagonanditscircumscribedcircle,illustrateanddescribeaprocessforconstructingahexagoninscribedinthecirclegivenbelow.

5. Modifyyourworkwiththehexagontoconstructanequilateraltriangleinscribedinthecirclegivenbelow.

6. Describehowyoumightconstructasquareinscribedinacircle.

9






7.2

READY Topic:Transformationoflines,connectinggeometryandalgebra.Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichisthepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethetransformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline.1. 2.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:

3.

4.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:

HG

H'G'


B'

A' B

A

A'

B' B

A

M'

P P'

M

10






7.2

SET Topic:Geometricconstructionswithcompassandstraightedge.

5.Constructaparallelogramgivensides!"and!"and∠ !"#.

6.Constructalineparallelto!"andthroughpointR.

7.GivensegmentAB showallpointsCsuchthatΔ ABC isanisoscelestriangle.8.GivensegmentAB showallpointsCsuchthatΔ ABC isarighttriangle.

Z

Y

X

11






7.2

GO Topic:Creatingexplicitandrecursiverulesforvisualpatterns9.Findanexplicitfunctionruleandarecursiverulefordotsinstepn.

Step1 Step2 Step3

10.Findanexplicitfunctionruleandarecursiveruleforsquaresinstepn.

Step1 Step2 Step3

Findanexplicitfunctionruleandarecursiveruleforthevaluesineachtable.

11. 12. 13.

n f(n)2 163 84 45 2

n f(n)1 -52 253 -1254 625

Step Value1 12 113 214 31

12






7. 3 Can You Get There From

Here?


Thetwoquadrilateralsshownbelow,quadrilateralABCDandquadrilateralQRSTarecongruent,withcorrespondingcongruentpartsmarkedinthediagrams.

Describeasequenceofrigid-motiontransformationsthatwillcarryquadrilateralABCDontoquadrilateralQRST.Beveryspecificindescribingthesequenceandtypesoftransformationsyouwillusesothatsomeoneelsecouldperformthesameseriesoftransformations.

CC

BY

Cal

sidy

rose

http

s://f

lic.k

r/p/

8veh

vb

13






7.3

READY Topic:Multipletransformations

Thegivenfiguresaretobeusedaspre-images.Performthestatedtransformationstoobtainanimage.Labelthecorrespondingpartsoftheimageinaccordancewiththepre-image.

1.ReflecttriangleABCovertheline! = !andlabeltheimageA’B’C’.

RotatetriangleA’B’C’1800counterclockwisearoundtheoriginandlabeltheimageA’’B’’C’’.

2.Reflectovertheline! = −!.

3.Reflectovery-axisandthen

Rotateclockwise900aroundP’.

X

ZY

N

M

LK

P

O


-10

-10

10

10

C

A

B

14






7.3

4.ReflectquadrilateralABCDovertheline

! = 2 + !andlabeltheimageA’B’C’D’.

RotatequadrilateralA’B’C’D’counter-clockwise900

around(-2,-3)asthecenterofrotationlabelthe

imageA’’B’’C’’D’’.

SET Topic:Findthesequenceoftransformations.

FindasequenceoftransformationsthatwillcarrytriangleRSTontotriangleR’S’T’.Clearlydescribethesequenceoftransformationsbeloweachgrid.

5.

6.

CD

AB

15






7.3

GO Topic:Graphingsystemsoffunctionsandmakingcomparisons.

Grapheachpairoffunctionsandmakeanobservationabouthowthefunctionscomparetooneanother.

7.! = !

! ! − 1! = −3! − 1

8.! = − !

! ! + 5! = !

! ! + 5

9.! = !

! ! + 2! = − !

! + 2

10.! = 2!! = −2!

-10

-10

10

10

-10

-10

10

10

-10

-10

10

10

-10

-10

10

10

16






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:

CC

BY

She

rrie

Tha

i

http

s://f

lic.k

r/p/

5ezx

Wj

17






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.

18






7. Giveninformation:__________ Arethetrianglescongruent?_________




19








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?

20






7.4

READY Topic:Correspondingpartsoffiguresandtransformations.Giventhefiguresineachsketchwithcongruentanglesandsidesmarked,firstlistthepartsofthefiguresthatcorrespond(Forexample,in#1,∠ ! ≅ ∠!)Thendetermineifareflectionoccurredaspartofthesequenceoftransformationsthatwasusedtocreatetheimage.

1.

Congruencies

∠ ! ≅ ∠ !

Reflected?YesorNo

2.

Congruencies

Reflected?YesorNo


T

SR

C

A

B

Z

W

X

Y

F

EH

G

21






7.4

SET Topic:TriangleCongruenceExplainwhetherornotthetrianglesarecongruent,similar,orneitherbasedonthemarkingsthatindicatecongruence.

3.

4.

5. 6.

7.

8.

Usethegivencongruencestatementtodrawandlabeltwotrianglesthathavethepropercorrespondingpartscongruenttooneanother.

9.∆ !"# ≅ ∆ !"# 10.∆ !"# ≅ ∆ !"#

22






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, ...

23






7. 5 Congruent Triangles

to the Rescue

A Practice Understanding Task

Part1

ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent:

Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertex

pointoftheanglemadeupbythetwocongruentsides,andthemidpointofthethirdside.

Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.

Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.

Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.

Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.

1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineof

symmetry?Ifso,whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabout

hisstatement?

2. WhatelsewouldZacneedtoknowaboutthecreaselinethroughinordertoknowthatitisa

lineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)

3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)

createstwocongruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASor

SSS—couldheusetosupportthisclaim?Describethesidesand/oranglesyouthinkare

congruent,andexplainhowyouknowtheyarecongruent.

4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes

thatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenot

formedbythetwocongruentsides)?

CC

BY

And

ers

Sand

berg

http

s://f

lic.k

r/p/

3GZ

ScG

24






5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoes

thatimplyaboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthis

line—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non-congruent

sideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthe

vertexangleformedbythetwocongruentsides.)

Part2

LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotational

symmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinitionyou

madesomeobservationsaboutsides,angles,anddiagonalsofvarioustypesofquadrilateralsbased

onyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbe

furtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZac

andSionedidintheirworkwithisoscelestriangles.

Pickoneofthefollowingquadrilateralstoexplore:

• Arectangleisaquadrilateralthatcontainsfourrightangles.

• Arhombusisaquadrilateralinwhichallsidesarecongruent.

• Asquareisbotharectangleandarhombus,thatis,itcontainsfourrightanglesandallsidesarecongruent

1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthe

quadrilateralA,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.

2. Basedon(1)yourdrawing,(2)thegivendefinitionofyourquadrilateral,and(3)information

aboutsidesandanglesthatyoucangatherbasedonlinesofreflectionandrotational

symmetry,listasmanypairsofcongruenttrianglesasyoucanfind.

3. Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused—ASA,SASorSSS—to

determinethatthetwotrianglesarecongruent,andexplainhowyouknowthattheangles

and/orsidesrequiredbythecriteriaarecongruent(seethefollowingchart).

25






CongruentTriangles

CriteriaUsed(ASA,SAS,SSS)

HowIknowthesidesand/oranglesrequiredbythecriteriaarecongruent

IfIsayΔRST≅ΔXYZ

basedonSSS

thenIneedtoexplain:

• howIknowthat

�

RS ≅ XY ,and• howIknowthat

�

ST ≅ YZ ,and• howIknowthat

�

TR ≅ ZX soIcanuseSSScriteriatosayΔRST≅ΔXYZ

4. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethe

congruenttrianglestojustifysomethingelseaboutthequadrilateral,suchas:

• thediagonalsbisecteachother

• thediagonalsarecongruent

• thediagonalsareperpendiculartoeachother

• thediagonalsbisecttheanglesofthequadrilateral

Pickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandtryto

writeanargumentthatwouldconvinceZacandSionethatthestatementistrue.

26




Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

7.5

READY Topic:Transformationsoflines,connectinggeometryandalgebra.Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichisthepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethetransformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline.1.

2.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:Useforproblems3thorugh5.

3.a.DescriptionofTransformation:b.Equationforpre-image:c.Equationforimage:4.Writeanequationforalinewiththesameslopethatgoesthroughtheorigin.5.WritetheequationofalineperpendiculartotheseandthoughthepointO’.

M

N

M'

N'


27





7.5

Afterworkingwiththeseequationsandseeingthetransformationsonthecoordinategraphitisgoodtimingtoconsidersimilarworkwithtables.6.Matchthetableofvaluesbelowwiththeproperfunctionrule.I II III IV V

x f(x)-1 160 141 122 10

x f(x)-1 140 121 102 8

x f(x)-1 120 101 82 6

x f(x)-1 100 81 62 4

x f(x)-1 80 61 42 2

A.! ! = −! ! − ! + ! D.! ! = −! ! + ! + ! B.! ! = −! ! − ! + !" E.! ! = −! ! + ! + !" C.! ! = −! ! − ! + ! SET Topic:UseTriangleCongruenceCriteriatojustifyconjectures.Ineachproblembelowtherearesometruestatementslisted.Fromthesestatementsaconjecture(aguess)aboutwhatmightbetruehasbeenmade.Usingthegivenstatementsandconjecturestatementcreateanargumentthatjustifiestheconjecture.

7.Truestatements: PointMisthemidpointof!"∠!"# ≅ ∠!"#!" ≅ !"

Conjecture:∠A ≅∠C a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

28





7.5

8.Truestatements∠ !"# ≅ ∠ !"#!" ≅ !"

Conjecture:!"bisects∠ !"#a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

9.Truestatements∆ !"#isa180°rotationof∆ !"#

Conjecture:∆ !"# ≅ ∆!"#a.Istheconjecturecorrect?b.Argumenttoproveyouareright:

GO Topic:Constructionswithcompassandstraightedge.10.Whydoweuseageometriccompasswhendoingconstructionsingeometry?

29





7.5

Performtheindicatedconstructionsusingacompassandstraightedge.11.Constructarhombus,usesegmentABasonesideandangleAasoneoftheangles.12.ConstructalineparalleltolinePRandthroughthepointN.13.ConstructanequilateraltrianglewithsegmentRSasoneside.14.Constructaregularhexagoninscribedinthecircleprovided.15.ConstructaparallelogramusingCDasonesideandCEastheotherside.16.BisectthelinesegmentLM. 17.BisecttheangelRST.

30






7. 6 Justifying Constructions

A Solidify Understanding Task

Compassandstraightedgeconstructionscanbejustifiedusingsuchtoolsas:

•thedefinitionsandpropertiesoftherigid-motiontransformations

•identifyingcorrespondingpartsofcongruenttriangles

•usingobservationsaboutsides,anglesanddiagonalsofspecialtypesofquadrilaterals

Studythestepsofthefollowingprocedureforconstructingananglebisector,andcompletethe

illustrationbasedonthedescriptionsofthesteps.

Steps Illustration Usingacompass,drawanarc(portionofacircle)thatintersectseachrayoftheangletobebisected,withthecenterofthearclocatedatthevertexoftheangle.

Withoutchangingthespanofthecompass,drawtwoarcsintheinterioroftheangle,withthecenterofthearcslocatedatthetwopointswherethefirstarcintersectedtheraysoftheangle.Withthestraightedge,drawarayfromthevertexoftheanglethroughthepointwherethelasttwoarcsintersect.

Explainindetailwhythisconstructionworks.Itmaybehelpfultoidentifysomecongruent

trianglesorafamiliarquadrilateralinthefinalillustration.Youmayalsowanttousedefinitionsor

propertiesoftherigid-motiontransformationsinyourexplanation.Bepreparedtoshareyour

explanationwithyourpeers.

CC

BY

TH

OR

http

s://f

lic.k

r/p/

9QK

xv

31






Studythestepsofthefollowingprocedureforconstructingalineperpendiculartoagivenline

throughagivenpoint,andcompletetheillustrationbasedonthedescriptionsofthesteps.

Steps IllustrationUsingacompass,drawanarc(portionofacircle)thatintersectsthegivenlineattwopoints,withthecenterofthearclocatedatthegivenpoint.

Withoutchangingthespanofthecompass,locateasecondpointontheothersideofthegivenline,bydrawingtwoarcsonthesamesideoftheline,withthecenterofthearcslocatedatthetwopointswherethefirstarcintersectedtheline.

Withthestraightedge,drawalinethroughthegivenpointandthepointwherethelasttwoarcsintersect.





32






Studythestepsofthefollowingprocedureforconstructingalineparalleltoagivenlinethrougha

givenpoint,andcompletetheillustrationbasedonthedescriptionsofthesteps.

Steps Illustration

Usingastraightedge,drawalinethroughthegivenpointtoformanarbitraryanglewiththegivenline.

Usingacompass,drawanarc(portionofacircle)thatintersectsbothraysoftheangleformed,withthecenterofthearclocatedatthepointwherethedrawnlineintersectsthegivenline.

Withoutchangingthespanofthecompass,drawasecondarconthesamesideofthedrawnline,centeredatthegivenpoint.Thesecondarcshouldbeaslongorlongerthanthefirstarc,andshouldintersectthedrawnline.

Setthespanofthecompasstomatchthedistancebetweenthetwopointswherethefirstarccrossesthetwolines.Withoutchangingthespanofthecompass,drawathirdarcthatintersectsthesecondarc,centeredatthepointwherethesecondarcintersectsthedrawnline.

Withthestraightedge,drawalinethroughthegivenpointandthepointwherethelasttwoarcsintersect.





33






7.6

READY Topic:Rotationalsymmetryinregularpolygonsandwithtransformations.1.Whatanglesofrotationalsymmetryarethereforaregularpentagon?2.Whatanglesofrotationalsymmetryarethereforaregularhexagon?3.Ifaregularpolygonhasanangleofrotationalsymmetrythatis400,howmanysidesdoesthepolygonhave?Oneachgivencoordinategridbelowperformtheindicatedtransformation.4.

ReflectpointPoverlinej.

5.

RotatepointP900clockwisearoundpointC.

j

P

C

P


34






7.6

SET Topic:UseTriangleCongruenceCriteriatojustifyconjectures.6.ConstructanisoscelestrianglethatincorporatesCD asoneofthesides.Constructthecircumscribedcirclearoundthetriangle.7.ConstructaregularhexagonthatincorporatesCD asoneofthesides.Constructthecircumscribedcirclearoundthehexagon.8.ConstructasquarethatincorporatesCD asoneofthesides.Constructthecircumscribedcirclearoundthesquare.

35






7.6

GO Topic:FindingDistanceandSlope.Foreachpairofgivencoordinatepointsfinddistancebetweenthemandfindtheslopeofthelinethatpassesthroughthem.Showallyourwork.9.(-2,8),(3,-4) 10.(-7,-3),(1,5))a.Slope:b.Distance:

a.Slope:b.Distance:

11.(3,7),(-5,9) 12.(1,-5)(-7,1)a.Slope:b.Distance:

a.Slope:b.Distance:

13.(-10,31)(20,11) 14.(16,-45)(-34,75)a.Slope:b.Distance:

a.Slope:b.Distance:

36

Module 7 - Mathematics Vision Project

Documents