Contents · 2016-10-05 · Lesson 7 Using Congruence to Prove Theorems . . . . . . . . . . . . . 46...

Contents

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Problem Solving Performance Task

Unit 1 Congruence, Proof, and Constructions . . . . . . . . . 4

Lesson 1 Transformations and Congruence . . . . . . . . . . . . . . . . . . 6

Lesson 2 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Lesson 3 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Lesson 4 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Lesson 5 Symmetry and Sequences of Transformations . . . . . . . 32

Lesson 6 Congruent Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Lesson 7 Using Congruence to Prove Theorems . . . . . . . . . . . . . 46

Lesson 8 Constructions of Lines and Angles . . . . . . . . . . . . . . . . 54

Lesson 9 Constructions of Polygons . . . . . . . . . . . . . . . . . . . . . . . 64

Unit 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Unit 1 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Unit 2 Similarity, Proof, and Trigonometry . . . . . . . . . . . 78

Lesson 10 Dilations and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Lesson 11 Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Lesson 12 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Lesson 13 Relationships between Trigonometric Functions . . . . 102

Lesson 14 Solving Problems with Right Triangles . . . . . . . . . 108

Lesson 15 Trigonometric Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Unit 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Unit 2 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Unit 3 Extending to Three Dimensions . . . . . . . . . . . . . . 130

Lesson 16 Circumference and Area of Circles . . . . . . . . . . . . 132

Lesson 17 Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . 140

Lesson 18 Volume Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Unit 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158


G.CO.1, G.CO.2, G.CO.6

G.CO.1, G.CO.2, G.CO.4, G.CO.5



G.CO.3, G.CO.5

G.CO.7, G.CO.8

G.CO.9, G.CO.10, G.CO.11

G.CO.12

G.CO.13

G.SRT.1a, G.SRT.1b, G.SRT.2

G.SRT.2, G.SRT.3, G.SRT.4, G.SRT.5

G.SRT.6

G.SRT.7

G.SRT.8, G.SRT.9, G.MG.1, G.MG.2, G.MG.3

G.SRT.10, G.SRT.11

G.GMD.1, G.MG.1

G.GMD.4, G.MG.1

G.GMD.1, G.GMD.3, G.MG.1

Common Core StandardsCommon Core Standards





G.CO.3, G.CO.5

G.CO.7, G.CO.8

G.CO.9, G.CO.10, G.CO.11

G.CO.12

G.CO.13



G.SRT.6

G.SRT.7


G.SRT.10, G.SRT.11

G.GMD.1, G.MG.1

G.GMD.4, G.MG.1


Common Core Standards





G.CO.3, G.CO.5

G.CO.7, G.CO.8

G.CO.9, G.CO.10, G.CO.11

G.CO.12

G.CO.13



G.SRT.6

G.SRT.7


G.SRT.10, G.SRT.11

G.GMD.1, G.MG.1

G.GMD.4, G.MG.1







G.CO.3, G.CO.5

G.CO.7, G.CO.8

G.CO.9, G.CO.10, G.CO.11

G.CO.12

G.CO.13



G.SRT.6

G.SRT.7


G.SRT.10, G.SRT.11

G.GMD.1, G.MG.1

G.GMD.4, G.MG.1


New York State Common Core Learning Standards for Mathematics





G.CO.3, G.CO.5

G.CO.7, G.CO.8

G.CO.9, G.CO.10, G.CO.11

G.CO.12

G.CO.13



G.SRT.6

G.SRT.7


G.SRT.10, G.SRT.11

G.GMD.1, G.MG.1

G.GMD.4, G.MG.1


Arizona's College and Career Ready Standards


HS.G.CO.1, G.CO.2, G.CO.4, G.CO.5



HS.G.CO.3, G.CO.5

HS.G.CO.7, G.CO.8

HS.G.CO.9, G.CO.10, G.CO.11

HS.G.CO.12

HS.G.CO.13

HS.G.SRT.1a, G.SRT.1b, G.SRT.2

HS.G.SRT.2, G.SRT.3, G.SRT.4, G.SRT.5

HS.G.SRT.6

HS.G.SRT.7

HS.G.SRT.8, G.SRT.9, G.MG.1, G.MG.2, G.MG.3

HS.G.SRT.10, G.SRT.11

HS.G.GMD.1, G.MG.1

HS.G.GMD.4, G.MG.1

HS.G.GMD.1, G.GMD.3, G.MG.1

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Unit 4 Connecting Algebra and Geometry through Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 164

Lesson 19 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . 166

Lesson 20 Distance in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Lesson 21 Dividing Line Segments . . . . . . . . . . . . . . . . . . . . . . . . 178

Lesson 22 Equations of Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . 184

Lesson 23 Using Coordinate Geometry to Prove Theorems . . . . 190

Unit 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196


Unit 5 Circles With and Without Coordinates . . . . . . . 202

Lesson 24 Circles and Line Segments . . . . . . . . . . . . . . . . . . . . . . 204

Lesson 25 Circles, Angles, and Arcs . . . . . . . . . . . . . . . . . . . . . . . 214

Lesson 26 Arc Lengths and Areas of Sectors . . . . . . . . . . . . . . . . 224

Lesson 27 Constructions with Circles . . . . . . . . . . . . . . . . . . . . . . 230

Lesson 28 Equations of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Lesson 29 Using Coordinates to Prove Theorems about Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Unit 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250


Unit 6 Applications of Probability . . . . . . . . . . . . . . . . . . . 258

Lesson 30 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Lesson 31 Modeling Probability . . . . . . . . . . . . . . . . . . . . . . . 268

Lesson 32 Permutations and Combinations . . . . . . . . . . . . . 278

Lesson 33 Calculating Probability . . . . . . . . . . . . . . . . . . . . . . . . . 284

Lesson 34 Conditional Probability . . . . . . . . . . . . . . . . . . . . . 294

Lesson 35 Using Probability to Make Decisions . . . . . . . . . . . . . . 304

Unit 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310


Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Formula Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Math Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

G.GPE.4, G.GPE.5

G.GPE.4, G.GPE.7

G.GPE.4, G.GPE.6

G.GPE.2

G.GPE.4

G.C.1, G.C.2, G.MG.1

G.C.2, G.C.3

G.C.5, G.MG.1

G.C.3, G.C.4, G.MG.1

G.GPE.1

G.GPE.4

S.CP.1

S.CP.4

S.CP.9

S.CP.2, S.CP.7, S.CP.8

S.CP.3, S.CP.5, S.CP.6, S.CP.8

S.MD.6, S.MD.7

Common Core StandardsCommon Core Standards

G.GPE.4, G.GPE.5

G.GPE.4, G.GPE.7

G.GPE.4, G.GPE.6

G.GPE.2

G.GPE.4

G.C.1, G.C.2, G.MG.1

G.C.2, G.C.3

G.C.5, G.MG.1

G.C.3, G.C.4, G.MG.1

G.GPE.1

G.GPE.4

S.CP.1

S.CP.4

S.CP.9



S.MD.6, S.MD.7


G.GPE.4, G.GPE.5

G.GPE.4, G.GPE.7

G.GPE.4, G.GPE.6

G.GPE.2

G.GPE.4

G.C.1, G.C.2, G.MG.1

G.C.2, G.C.3

G.C.5, G.MG.1

G.C.3, G.C.4, G.MG.1

G.GPE.1

G.GPE.4

S.CP.1

S.CP.4

S.CP.9



S.MD.6, S.MD.7


G.GPE.4, G.GPE.5

G.GPE.4, G.GPE.7

G.GPE.4, G.GPE.6

G.GPE.2

G.GPE.4

G.C.1, G.C.2, G.MG.1

G.C.2, G.C.3

G.C.5, G.MG.1

G.C.3, G.C.4, G.MG.1

G.GPE.1

G.GPE.4

S.CP.1

S.CP.4

S.CP.9



S.MD.6, S.MD.7

New York State Common Core Learning Standards for Mathematics

G.GPE.4, G.GPE.5

G.GPE.4, G.GPE.7

G.GPE.4, G.GPE.6

G.GPE.2

G.GPE.4

G.C.1, G.C.2, G.MG.1

G.C.2, G.C.3

G.C.5, G.MG.1

G.C.3, G.C.4, G.MG.1

G.GPE.1

G.GPE.4

S.CP.1

S.CP.4

S.CP.9



S.MD.6, S.MD.7

Arizona's College and Career Ready Standards

HS.G.GPE.4, G.GPE.5

HS.G.GPE.4, G.GPE.7

HS.G.GPE.4, G.GPE.6

HS.G.GPE.2

HS.G.GPE.4

HS.G.C.1, G.C.2, G.MG.1

HS.G.C.2, G.C.3

HS.G.C.5, G.MG.1

HS.G.C.3, G.C.4, G.MG.1

HS.G.GPE.1

HS.G.GPE.4

HS.S.CP.1

HS.S.CP.4

HS.S.CP.9

HS.S.CP.2, S.CP.7, S.CP.8

HS.S.CP.3, S.CP.5, S.CP.6, S.CP.8

HS.S.MD.6, S.MD.7


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UNDERSTAND Twotrianglesarecongruentifalloftheircorrespondinganglesarecongruentandalloftheircorrespondingsidesarecongruent .However,youdonotneedtoknowthemeasuresofeverysideandangletoshowthattwotrianglesarecongruent .

nABCisformedfromthreelinesegments:___

AB ,___

BC ,and___

AC .Supposethosesegmentswerepulledapartandusedtobuildanothertriangle,suchasnA9B9C9shown .ThisnewtrianglecouldbeformedbyreflectingnABCoveraverticalline .A reflectionisarigidmotion,sonA9B9C9mustbecongruenttonABC. Infact,anytrianglebuiltwiththesesegmentscouldbeproducedbyperformingrigidmotionsonnABC,soallsuchtriangleswouldbecongruenttonABC .

So,knowingallofthesidelengthsoftwotrianglesisenoughinformationtodetermineiftheyarecongruent .

Side-Side-Side (SSS) Postulate: Ifthreesidesofonetrianglearecongruenttothreesidesofanothertriangle,thenthetrianglesarecongruent .

UNDERSTAND YoucanalsousetheSide-Angle-Side(SAS)Postulatetoprovethattwotrianglesarecongruent .

Side-Angle-Side (SAS) Postulate: Iftwosidesandtheincludedangleofonetrianglearecongruenttotwosidesandtheincludedangleofanothertriangle,thenthetrianglesarecongruent .

LookatnDEFonthecoordinateplane .Applyingrigidmotionstotwoofitssidesproducednewlinesegments .Sides

___DE and

__EF

weretranslated7unitstotherighttoform___

GH and__

HI .___

DE and__

EF wererotated180°abouttheorigintoform

__JK and

__KL .

___DE and

__EF

werereflectedoverthex-axistoform____

MN and___

NO .

Eachrigidmotionpreservedthelengthsofthesegmentsaswellastheanglebetweenthem .Inallthreeimages,onlyonesegmentcanbedrawntocompleteeachtriangle .Thoselinesegments,shownasdottedlines,arecongruentto

___DF .

So,nDEFnGHInJKLnMNO. Thesymbolmeans"iscongruentto ."

Congruence with Multiple Sides

C� B�B

A A�

C

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

E

D

M

N

F

O L

K

J

H

GI

LESSON

6 Congruent Triangles

38 Unit 1: Congruence, Proof, and Constructions


Lesson 6: Congruent Triangles 39

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ThecoordinateplaneontherightshowsnABCandnDEF.

UsetheSASPostulatetoshowthatthetrianglesarecongruent .Then,identifyrigidmotionsthatcouldtransformnABC intonDEF.

Makeaplan .

Eachtrianglehasonehorizontalsideandoneverticalsidethatintersect,sobotharerighttriangles .Sinceallrightanglesarecongruent,thetriangleshaveatleastonepairofcorrespondingcongruentangles .ToprovethetrianglescongruentbytheSASPostulate,findthelengthsoftheadjacentsides(legs)thatformtherightangles .

IdentifyrigidmotionsthatcantransformnABCtonDEF.

Studytheshapesofthetriangles .

▸___

AC correspondsto___

DF andisparalleltoit .VertexBisabove

___AC ontheleft,

whilevertexE liesbelow___

DF andisalsoontheleft .n ABCcouldbereflectedoverthex-axis .Aftersuchareflection,n A9B9C9wouldhaveverticesA9(23,21),B9(23, 24),andC9(2,21) .TotransformthisimagetonDEF, translateit3unitstotheleft .

3

1

IfnABChadinsteadbeenrotated90°andthentranslateddown1unit,wouldtheresultingimagebecongruenttonABC?Howcouldyouproveyouranswer?

Findandcomparethelengthsofthecorrespondinglegs .

Counttheunitstofindthelengthofeachleg .

AB53 DE53

AC55 DF 5 5

___

AB ___

DE and___

AC ___

DF

▸Thetriangleshavetwopairsofcorrespondingcongruentsides,andtheirincludedanglesarecongruentrightangles .So,accordingtotheSASPostulate,nABC nDEF .

2

y

–2

–4

–6

–2–4–6 2 4 60

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B

A

D

E

C

F

DISCUSS


40 Unit : Congruence, Proof, and Constructions

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UNDERSTAND AthirdmethodforprovingthattrianglesarecongruentistheAngle-Side-AngleTheorem .

Angle-Side-Angle (ASA) Theorem: Iftwoanglesandtheincludedsideofonetrianglearecongruenttotwoanglesandtheincludedsideofanothertriangle,thenthetrianglesarecongruent .

Lookatthecoordinateplanesbelow .TriangleGHJisshownontheleft .Ontheright,sides__

HJ and

___GJ ofthetrianglehavebeenreplacedbyrays .

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

G

JH

G

H

Thecoordinateplaneontherightofthisparagraphshowsthreetransformationsof

___GH andtheraysextendingfrompointsGandH. __

KL isatranslationof___

GH 6unitstotheright .___

ON isa180°rotationof

___GH .

___RQ isareflectionof

___GH overthex-axis .

Eachofthoserigidmotionshascarriedtheangle-side-anglecombinationtoanewlocation .Thesegments

__KL ,

___ON ,and

___RQ are

congruentto___

GH .Rigidmotionsalsopreservedtheanglesformedbythesegmentandeachray .

Thecoordinateplaneonthelowerrightshowsthetrianglesformedbyextendingtheraysuntiltheyintersect .

Ineachcase,therayscanonlyintersectatonepointandthuscanformonlyonetriangle .EachofthesetrianglesiscongruenttonGHJ.

Congruence with Multiple Angles

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

G

H

R

Q

O

N

K

L

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

G

J

S

R

H

Q N

O

P

K

M L

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TrianglesPQRandSTUareshownonthecoordinateplaneontheright .

Giventhat/P/Sand/R/U,provethatthetrianglesarecongruent .ThenidentifyrigidmotionsthatcantransformnPQRinto nSTU.

Makeaplan .

Youalreadyknowthattwopairsofcorrespondinganglesarecongruent .Ifyoucanshowthattheincludedsidesarethesamelength,thenthetrianglesarecongruentbytheASATheorem .

IdentifyrigidmotionsthatcouldtransformnPQRintonSTU.

Studytheshapesofthetriangles .

Side___

PR ishorizontal,andcorrespondingside

___SU isvertical .ItappearsthatnPQR

wasrotated90°counterclockwise .If thatrotationwerearoundtheorigin,nP9Q9R9wouldhaveverticesatP9(22, 1),Q9(26, 3),andR9(22,6) .TotransformthisimagetonSTU,translateit2unitsdown .

▸TriangleSTUcanbeproducedbyrotatingnPQR90°counterclockwiseandthentranslatingtheimagedown2units .

3

1

Findthelengthsoftheincludedsides .

InnPQR,theincludedsideof/Pand/Ris

___PR .Findthelengthof

___PR by

countingunits .

PR 5 5

InnSTU,theincludedsidefor/Sand/Uis

___SU .Findthelengthof

___SU by

countingunits .

SU55

▸Sincetwopairsofcorrespondinganglesandtheincludedsidesarecongruent,nPQRnSTUbytheASATheorem .

2

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

Q

PT

S

R

U


IntrianglesABCandDEF,/A 5/D520,/B 5/E560,and/C 5/F5100 .AretrianglesABCandDEFcongruent?

DISCUSS


msemrad

Delete

hkounne

Text Box

delete stray character

42 Unit : Congruence, Proof, and Constructions

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EXAMPLE A ShowthatnTUViscongruenttonXYZ.

Makeaplan .

TousetheSSSPostulate,youneedtoshowthateachsideonnXYZhasacorrespondingcongruentsideonnTUV .Todoso,showthateachsideofnXYZisarigid-motiontransformationofasideofnTUV.

Followthesameprocessfortheothertwopairsofcorrespondingsides .

Theendpointsof___

YZ and___

UV havethesamey-coordinatesbutoppositex-coordinates .

Theendpointsof___

XZ and___

TV alsohavethesamey-coordinatesbutoppositex-coordinates .

So,___

YZ isareflectionof___

UV overthey-axis,and

___XZ isareflectionof

___TV overthey-axis .

3

1

Comparetheendpointsof___

XY and___

TU .

PointsTandXhavethesamey-coordinatesbutoppositex-coordinates .ThesameistrueforpointsUandY .Thisindicatesthat___

TU canbereflectedoverthey-axistoform___XY .Asaresult,youknowthat

___XY

___TU .

2

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

UT

Z

X

V

Y

Drawaconclusion .

▸SinceeachsideofnXYZisareflectionofthecorrespondingsideofnTUV,thecorrespondingsidesofthetrianglesarecongruent .AccordingtotheSSSPostulate,nTUVnXYZ.

4

Howdoes/Zcompareto/V?Couldoneanglehaveagreatermeasurethantheother?


DISCUSS



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EXAMPLE B Ianisstudyingwingdesignsforairplanes .Hecomparestwowingswhosetriangularcrosssectionsbothcontainone20°angle,anadjacentsidethatmeasures6feet,andanon-adjacentsidethatmeasures3feet .Determineifthetriangularcrosssectionsareidentical .

6

63

3

20°20°

Makeaplan .

Identicaltrianglesarecongruent,sorigidmotionsshouldcarryoneofthefiguresontotheother .Transformoneofthefiguressothatknowncongruentpartslineup,andcomparetheotherparts .

Re-aligntheangles .

Theanglesarenolongeraligned,soreflecttheimageverticallytobringthembackintoalignment .

20°6

3

3

1

Attempttoaligntheangleandadjacent side .

Theanglesarealreadyaligned,butthe6-footsidesarenot .Rotatethesecondtriangle20°counterclockwisesothatits6-footsideisalsohorizontal .

6

320°

2

Translatetheimageontotheothertriangle .

20°

6

33

Theangleandadjacentsidearealigned,buttheremainingsidesandanglesdonot match .

▸Thecrosssectionsarenotcongruent .

4

Iftwotriangleshavetwopairsofcongruentsidesandonepairofcongruentangles,canyouprovethatthetrianglesarecongruent?

DISCUSS


Practice

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Practice

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Use the coordinate plane below for questions 1–4.

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

L O M

N

1. /LON and/MONbothmeasure degrees .

2. LO 5OM 5 units

3. nLONandnMONshareside .

4. TrianglesLONandMONarecongruentbythe Postulate .

If two triangles share a side, that line segment is the same length in both triangles.

HINT

Choose the best answer.

5. WhichpairofrigidmotionsshowsthatnABCandnA9B9C9arecongruent?

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

xA

B

C

A�

B�

C�

A. reflectionacrossthex-axisfollowedbyatranslationof3unitsup

B. reflectionacrossthex-axisfollowedbyatranslationof3unitsdown

C. rotationof180°abouttheoriginfollowedbyareflectionoverthey-axis

D. rotationof90°counterclockwiseabouttheoriginfollowedbyatranslationof4unitsdown




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6. ThecoordinateplaneontherightshowsisoscelesrighttrianglesHIJandKLM .

UsetheASAPostulatetoprovethatnHIJandnKLMarecongruent .IdentifyrigidmotionsthatcouldtransformnHIJintonKLM.

7. TriangleABCwasreflectedhorizontally,reflectedvertically,andthentranslatedtoformtriangleA9B9C9 .Identifythelengthsandanglemeasuresbelow .

A9B95

AC5

m/B95

m/A5

Use the following information for questions 8 and 9.

RighttriangleNOPhasverticesN(1,1),O(1,5),andP(4,1) .RighttriangleQRShasverticesQ(24,21),R(24,25),andS(21,21) .

8. SKETCH Sketchbothtrianglesonthecoordinateplane .

9. PROVE ProvethatnNOPandnQRSarecongruent .

A

12

5118°

B

C

A�15

545°

B�

C�

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

y

–2

–4

–6

–2–4–6 2 4 60

2

4

6

x

H

I

K M

L

J


Contents · 2016-10-05 · Lesson 7 Using Congruence to Prove Theorems . . . . . . . . . . . . . 46...

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Transcript of Contents · 2016-10-05 · Lesson 7 Using Congruence to Prove Theorems . . . . . . . . . . . . . 46...