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Extra Practice Course 3B 57

Name:      Date:   

CHAPTER

9 Congruence and Similarity

Lesson 9.1 Understanding and Applying Congruent Figures

Name the figures that are congruent. Name the corresponding congruent line segments and angles.

1. ABC is an isosceles triangle and M is the midpoint of BC.

A

M CB

2. ABCD is a parallelogram and AP 5 CQ.

A BP

D Q C

Solve. Show your work.

3. ABCD is a kite, whose diagonals intersect at P. Name all possible pairs of figures that are congruent. For each pair, name the corresponding congruent line segments and angles.

B

CP

D

A

MIF_ExtraPractice C3_Ch09.indd 57 3/30/12 12:34 AM

Name:      Date:   

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58 Chapter 9 Lesson 9.1

Solve. Show your work.

4. ABCD is a rhombus, whose diagonals intersect at P. Explain, using a test for congruent triangles, why PAB PCD.

B C

A D

P

5. ABCDEF is a regular hexagon with diagonals AC, AD, and AE as shown. It is given that AC and AE are congruent diagonals.

B C

A D

EF

a) Name two pairs of congruent triangles. For each pair, justify the congruency with a test for congruent triangles.

b) Name a pair of congruent quadrilaterals.

MIF_ExtraPractice C3_Ch09.indd 58 3/30/12 12:34 AM

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Extra Practice Course 3B 59

Solve. Show your work.

6. In the diagram, ABC DEC. Find the values of u, v, w, x, y, and z.

E

B C10 in.

67.4° v°

w°

u°

26 in.

y cm

x cm

z cm

D

7. ABCDE QRSTP. Find the values of u, v, w, x, and y.

25 cm36 cm

20 cm

30 cm

69°

v°

(6x � 2y)cm

B

A

C

D

E

w – 2

S

P

Q R

18 cm

T(8x � 3y) cm

51°

u° v°

MIF_ExtraPractice C3_Ch09.indd 59 3/30/12 12:34 AM

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60 Chapter 9 Lesson 9.160 Chapter 9 Lesson 9.1

Solve. Show your work.

8. In the diagram, ABDE is a parallelogram. mACB 5 mDFE 5 90°.

a) Justify that ABC DEF with a test for congruent triangles.

b) Write the congruence statement for a quadrilateral that is congruent to quadrilateral ABDF.

c) Find the length of each side of the quadrilateral you named in b).

9. In the diagram, ABCD is a rhombus and CDE 5 ADE.

a) Identify two pairs of congruent triangles.

b) For each pair in a), determine which congruence test proves that the triangles are congruent. Explain.

B C

E

DA

EF

A

BC D4 ft

5 ft

3 ft

MIF_ExtraPractice C3_Ch09.indd 60 3/30/12 12:34 AM

176 Answers

c) Reflection about the line y 5 24

d) Dilation of scale factor 32

about center (1, 21)

6. a) There is no translation that maps the rhombus onto itself.

b) i) Reflection about the diagonal AC.

ii) Reflection about the diagonal BD.

c) i) Rotation of 1808 (clockwise or counterclockwise about the center of the rhombus (the center is the point where the diagonals cut)

ii) Rotation of 3608 (clockwise or counterclockwise) about the center of the rhombus.

d) Dilation of scale factor 1 or 21 with the centre of the rhombus as the center of dilation.

Brain@Work 1. A

Q

Q�

P�

P

B

M

(1) Draw the bisector of PQ and label it AB. AB is then the line of reflection.

(2) Draw a line through P9 perpendicular to PQ to cut AB at M. On this line, mark the point Q9 with MQ9 5 MP9. Q9 is then the image of Q in the line of reflection.

2. Transformation

D is mapped onto C

Rotation of 90º counterclockwise about the origin

C is mapped onto B

Reflection about the y-axis

B is mapped onto A

Translation of 8 units to the left.

Chapter 9

Lesson 9.1 1. ABM ACM

AB 5 AC, AM 5 AM, BM 5 CM mABM 5 mACM, mAMB 5 mAMC,

and mBAM 5 mCAM

2. ADQP CBPQ AD 5 CB, DQ 5 BP, QP 5 PQ, AP 5 CQ mPAD 5 mQCB, mADQ 5 mCBP, mDQP 5 mBPQ, and mAPQ 5 mCQP 3. 1st pair: ABC ADC AB 5 AD, AC 5 AC, BC 5 DC mCAB 5 mCAD, mABC 5 mADC,

and mACB 5 mACD 2nd pair: ABP ADP AB 5 AD, AP 5 AP, BP 5 DP mPAB 5 mPAD, mABP 5 mADP, and mAPB 5 mAPD 3rd pair: CBP CDP CB 5 CD, BP 5 DP, CP 5 CP mCBP 5 mCDP, mCPB 5 mCPD,

mBCP 5 mDCP 4. In PAB and PCD, AB 5 CD [S] (opposite sides of

rhombus) PA 5 PC [S] (diagonals of rhombus

bisect each other) PB 5 PD [S] (diagonals of rhombus

bisect each other) PAB PCD [SSS]

5. a) 1st pair: ABC AFE AB 5 AF [S] (sides of regular

polygon) BC 5 FE [S] (sides of regular

polygon) AC 5 AE [S] (given) ABC AFE [SSS] 2nd pair: ACD AED AC 5 AE [S] (given) CD 5 ED [S] (sides of regular

polygon) AD 5 AD [S] (common side) ACD AED [SSS]

b) ABCD AFED

MIF_ExtraPractice_C3_Ch07-11_Ans.indd 176 3/30/12 6:06 PM

Extra Practice Course 3B 177

6. mACB 5 mDCE 5 u° u 1 u 5 180 u 5 90 mDEC 5 mABC 5 67.4° v° 5 180° 2 90° 2 67.4° v 5 22.6 w° 5 180° 2 67.4° w 5 112.6 x cm 5 AB 5 DE 5 26 cm x 5 26 CE 5 CB 5 10 in.

In DCE, y 5 26 102 2�

5 24 (Z 1 10) cm 5 AC 5 DC 5 24 cm z cm 5 14 cm z 5 14 So, u 5 90, v 5 22.6, w 5 112.6, x 5 26, y 5 24, and z 5 14.

7. u° 5 mPQR 5 mEAB 5 69° u 5 69 v° 5 mEBA 5 mPRQ 5 180° 2 mPRQ 2 mPRQ 5 60° ( sum of ) v 5 60 (w 2 2) cm 5 ST 5 CD 5 20 cm w cm 5 22 cm w 5 22 (6x 1 2y) cm 5 BC 5 RS 5 18 cm 6x 1 2y 5 18 i.e. 3x 1 y 5 9 — Equation 1 (8x 1 3y) cm 5 PT 5 ED 5 25 cm 8x 1 3y 5 25 — Equation 2 Multiply Equation 1 by 3: 9x 1 3y 5 27 — Equation 3 Subtract Equation 2 from Equation 3: x 5 2 Substitute x 5 2 into Equation 1: 6(2) 1 2y 5 18 2y 5 18 2 12 2y 5 6 y 5 3

8. a) In ABC and DEF: mACB 5 mDFE 5 90° [R] (given) AB 5 DE [H] (opposite sides of

parallelogram) AC 5 DF [S] ( distance

between sides) ABC DEF [RHS]

b) DEAC ABDF c) DE 5 BA 5 5 ft EA 5 DB 5 3 1 4 5 7 ft

AC 5 5 32 2�

5 4 ft CD 5 4 ft (given)

9. a) 1st pair: ABC ADC 2nd pair: ADE CDE b) In ABC and ADC, AB 5 AD [S] (sides of rhombus) BC 5 DC [S] (sides of rhombus) AC 5 AC [S] (common side) ABC ADC [SSS] In ADE and CDE, AD 5 CD [S] (sides of rhombus) mADE 5 mCDE [A] (given) DE 5 DE [S] (common side) ADE CDE [SAS]

Lesson 9.2 1.

B D

60°

u°

w° x°

y°v°

In B, u° 5 v°

5 ° °180 602� (base s of isosceles )

5 60° In D, w° 5 x° 5 y° 5 60° (s of an equilateral ) At least two pairs of corresponding angles

have equal measures. So, B D. 2.

A E

100°

80°a° b°

d°

f °

e°c°

In A, c° 5 180° 2 100° (property of rhombus) 5 80° a° 5 c° 5 80° (property of rhombus) b° 5 100° (property of rhombus)

MIF_ExtraPractice_C3_Ch07-11_Ans.indd 177 3/30/12 3:06 PM