contents · 2016-10-05 · lesson 7 using congruence to prove theorems . . . . . . . . . . . . . 46...
TRANSCRIPT
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
Unit 3 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
G.CO.1, G.CO.2, G.CO.6
G.CO.1, G.CO.2, G.CO.4, G.CO.5
G.CO.1, G.CO.2, G.CO.4, G.CO.5
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.1, G.CO.2, G.CO.6
G.CO.1, G.CO.2, G.CO.4, G.CO.5
G.CO.1, G.CO.2, G.CO.4, G.CO.5
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 Standards
G.CO.1, G.CO.2, G.CO.6
G.CO.1, G.CO.2, G.CO.4, G.CO.5
G.CO.1, G.CO.2, G.CO.4, G.CO.5
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 Standards
G.CO.1, G.CO.2, G.CO.6
G.CO.1, G.CO.2, G.CO.4, G.CO.5
G.CO.1, G.CO.2, G.CO.4, G.CO.5
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
New York State Common Core Learning Standards for Mathematics
G.CO.1, G.CO.2, G.CO.6
G.CO.1, G.CO.2, G.CO.4, G.CO.5
G.CO.1, G.CO.2, G.CO.4, G.CO.5
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
Arizona's College and Career Ready Standards
G.CO.1, G.CO.2, G.CO.6
HS.G.CO.1, G.CO.2, G.CO.4, G.CO.5
HS.G.CO.1, G.CO.2, G.CO.4, G.CO.5
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 4 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
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 5 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
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
Unit 6 Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
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.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 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.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 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.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
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.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
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
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–2–4–6 2 4 60
2
4
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x
E
D
M
N
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GI
LESSON
6 Congruent Triangles
38 Unit 1: Congruence, Proof, and Constructions
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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
2
4
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x
B
A
D
E
C
F
DISCUSS
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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
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G
H
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L
y
–2
–4
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–2–4–6 2 4 60
2
4
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G
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40 Unit 1: Congruence, Proof, and Constructions
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Lesson 6: Congruent Triangles 41
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v
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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
Lesson 6: Congruent Triangles 41
IntrianglesABCandDEF,/A 5/D520,/B 5/E560,and/C 5/F5100 .AretrianglesABCandDEFcongruent?
DISCUSS
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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?
42 Unit 1: Congruence, Proof, and Constructions
DISCUSS
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Lesson 6: Congruent Triangles 43
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Lesson 6: Congruent Triangles 43
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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
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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
44 Unit 1: Congruence, Proof, and Constructions
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Lesson 6: Congruent Triangles 45
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Lesson 6: Congruent Triangles 45
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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
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