Chapter 9 Parallel LInes Answer Key - mvb-math

Chapter 9 Parallel Lines Name __________________________________________________________

Class ______________

Chapter 9-1 Proving Lines Parallel Chapter 9-2 Properties of Parallel Lines

179

Date ______________

Version A [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] In 1–2, use the figure given below.

4. Given: l m and r is a transversal.

E c d

1 2

A

l

B

4 3 C

a

D

b

m

r

F

1. If m1 2x 40 and m4 4x 10, find the value of x so that AB y CD. (1) 10 ✔ (2) 25

If c 103, what is the value of a b? ✔ (1) 154

(3) 180

(2) 177

(4) 206

5. In the given figure, if m1 m2, what is the value of x?

(3) 45

40°

(4) 50 2. If m2 x and m3 3x 12, find the value of x so that AB y CD.

x

m1 m2

35°

m3

(1) 40

m4

✔ (2) 48

(1) 50

✔ (3) 105

(3) 52

(2) 75

(4) 140

(4) 180 3. In the given figure, l m. Find the measure of x.

6. In the given figure, if m1 ⊥ m2 and a b, which of the following statements is not true?

r

a l

70°

35

°

x

m

b

m3 m2

m1

(1) 105 (2) 110 (3) 135 ✔ (4) 145

Copyright © Amsco School Publications, Inc.

(1) m2 m3 (2) m1 ⊥ m3 (3) A line parallel to m1 must be perpendicular to m3. ✔ (4) A line that intersects m1 must also intersect m2.

180

Chapter 9 Parallel Lines

Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] b

7. a. In the given figure, c d and lines a and b are transversals with line a bisecting QRS. If mx 48, what is my?

a

Answer: my 24

Q

S

24°

c

24°

24° R 48°

x

24° 132°

d

y

E

b. In the given figure, mBAC 100, mCFG 130, AB y CD, and CE y FG. Find mDCE.

B

24°

G

D

Answer: mDCE 50 50° 100° 80° 50° A C

130° F

8. Given: BC > CA, m6 m3 C 3 1 2 4

Prove: l m

8 7 B

Proof:

l

6 5 A

m

Statements

Reasons

1. BC > CA

1. Given.

2. 7 6

2. Isosceles triangle theorem.

3. m7 m6

3. Definition of congruent angles.

4. m6 m3

4. Given.

5. m7 m3

5. Transitive property.

6. 7 3

6. Definition of congruent angles.

7. l m

7. If two coplanar lines are cut by a transversal so that the corresponding angles are , then the two lines are .



Class ______________

Chapter 9-1 Proving Lines Parallel Chapter 9-2 Properties of Parallel Lines

181

Date ______________

Version B [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. In the given figure, which information would not guarantee that two lines are parallel?

23

4 5

1 d

4. In the figure given, if l m, what is the measure of x?

a

x

l

b

2x

c

m n

✔ (1) 4 1

(1) 15°

(3) 45°

(2) 3 5

(2) 30°

✔ (4) 60°

(3) 2 4

5. In the figure given, lines m and p are parallel and BD bisects ABC. What is the measure of x?

(4) a ⊥ d and b ⊥ d 2. In the given figure, if g r and b 93, what is the value of c d?

D

m

C 110°

x

s p

a b

g

c

r

d

A

B

✔ (1) 55°

(3) 65°

(2) 60°

(4) 70°

6. In the given figure, lines a and b are not parallel. Which of the following could not be the value of x? (1) 173

(3) 175

✔ (2) 174

(4) 180 103°

3. If lines m and n are parallel and are intersected by transversal y, what is the sum of the measures of the interior angles on the same side of line y?

a

x°

b

(1) 90° ✔ (2) 180° (3) 270° (4) 360°


c

(1) 76

(3) 78

✔ (2) 77

(4) 79

182


Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] g

g

7. a. In the given figure, AB y CD, AE ' CB, mBCD 2x, and

B

A

mBAE 3x. Find the value of x.

3x

Answer: x 18°

E 2x

Solution: Alternate interior angles formed by cutting parallel lines are congruent, so mABE 2x.

D

C

2x 1 3x 5 90 5x 5 90 x 5 18 b. In the given figure, AB y DE, BC y EF, mBAD 33, and mBCD 58. Find mDEF.

B E

Answer: mDEF 89 Solution: Corresponding angles formed by cutting parallel lines are congruent, so mEDF 33 and mEFD 58.

A

D

C

F

33 1 58 1 m/DEF 5 180 m/DEF 5 89 8. Given: 3 1 and 2 3 g

E

g

Prove: EG y DH

3

A

B D

C

Proof:

G

Statements

Reasons

1. 2 3

1. Given.

g

2 F

1 H

g

2. AB y CD

2. If two coplanar lines are cut by a transversal so that the corresponding angles are , then the two lines are .

3. 3 1

3. Given.

4. 2 1


g

g

5. EG y DH

5. If two coplanar lines are cut by a transversal so that the corresponding angles are , then the two lines are . Copyright © Amsco School Publications, Inc.


Class ______________

Chapter 9-3 Parallel Lines in the Coordinate Plane

183

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] g

g

g

1. If AB y CD and the slope of AB is 12 , then the slope of g

CD is (1) 2 (2) 212 ✔ (3) 12 (4) 2 2. Find the equation of the line through (1, 2) and parallel to the line with equation y 7x 2. (1) y 217x 2 13 7 (2) y 217x 1 15 7 ✔ (3) y 7x 9 (4) y 7x 9

3. Which is the equation of the line that is parallel to the x-axis and passes through the point (1, 4)? (1) x 1 (2) y 1 (3) x 4 ✔ (4) y 4


4. Which is the equation of a line passing through (2, 5) and parallel to the line whose equation is y 3x 2? ✔ (1) y 3x 11 (2) y 3x 5 (3) y x 7 (4) y 3x 1 5. What is the slope of a line parallel to AB if A(4, 3) and B(7, 1)? (1) 11 2 2 ✔ (2) 11 2 (3) 211

(4) 211 2 6. The lines x 2y 5 and 4x + ky 5 are parallel for which value of k? (1) 4 (2) 1 (3) 212 ✔ (4) 8

184


Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 7. The coordinates of quadrilateral ABCD are A(5, 4), B(1, 2), C(2, 3), and D(4, 1). Show that both pairs of opposite sides are parallel. The slopes of opposite sides are equal: 22 2 (24)

3 2 (22) 221 5 5 1 2 (24) 5 24 2 (25) 5

Slope of AB 5 1 2 (25) 5 26 5 13

Slope of BC 5

321 2 1 Slope of CD 5 2 2 (24) 5 6 5 3

Slope of DA

5

8. Trapezoid PQRS has parallel bases PQ and SR. The coordinates of the vertices are P(0, 0), Q(k, 5), R(7, 1), and S(k, 3). a. Express the slope of PQ in terms of k. Answer: k5 b. Express the slope of SR in terms of k. 2 Answer: k22 2 7 or 7 2 k

c. Write an equation that can be used to solve for k and solve this equation for k. 5 k

2 5 72 k 2k 5 5(7 2 k)

2k 5 35 2 5k 7k 5 35 k 5 5 Answer



Class ______________

Chapter 9-4 The Sum of the Measures of the Angles of a Triangle

185

Date ______________


PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the measure of the angles of a triangle are in the ratio 3 : 5 : 7, then what is the measure of the largest angle?

4. In the given figure, l m, mBAD 49, AB bisects CAD, and CB bisects ACE. D

A

(1) 36°

l

(2) 60° x° B

✔ (3) 84° (4) 96° C

2. In the given figure, two angle measures are shown.

m

E

Find the value of x. B

144° D

A

x°

62° C

(1) 49

✔ (3) 90

(2) 82

(4) 98

5. If the degree measures of three angles of a triangle are represented by x 20, 5x 50, and 9x 40, the triangle must be

What is the value of x?

(1) right

(1) 36

(2) isosceles

(2) 82

(3) acute

✔ (3) 98 (4) 102 3. In triangle ABC, the measure of A is twice the measure of B, and an exterior angle at vertex C measures 117°. What is the measure of A?

✔ (4) scalene 6. In the given figure, l m, mDAB 30, mADB 125, and ABC is a right triangle. C x

l

D 125°

(1) 39°

30°

A

✔ (2) 78°

B

(3) 102°

What is the measure of x?

(4) 117°

(1) 25°

(3) 105°

✔ (2) 65°

(4) 155°


m

186


Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 7. In the given figure, mD 45, mA 35, and BC y DE. Find the degree measures of the angles numbered 1 to 6.

D

Answer: m1 80, m2 100, m3 80, m4 135, m5 100, m6 45 Solution: Since BC y DE, D and 6 are congruent corresponding angles. Therefore, m6 mD 45.

45°

B 4 6

2 1 E

5 3 C

35° A

F

The sum of the degree measures in a triangle is 180, so: m/A 1 m/6 1 m/5 5 180

m/A 1 m/D 1 m/2 5 180

35 1 45 1 m/5 5 180

35 1 45 1 m/2 5 180

m/5 5 100

m/2 5 100

1 and 2, 3 and 5, and 4 and 6 form linear pairs, so: m/1 1 m/2 5 180

m/3 1 m/5 5 180

m/4 1 m/6 5 180

m/1 1 100 5 180

m/3 1 100 5 180

m/4 1 45 5 180

m/1 5 80

m/3 5 80

m/4 5 135 S

8. In RST, the bisectors of the angles meet at point P. If mRTS 60, mRPT 125, and mRPS 120, find the degree measures of the angles numbered 1 to 4.

32

Answer: m1 115, m2 35, m3 35, m4 25 Solution: TP bisects RTS, so mRTP mSTP 12 (60) 30.

1 P

The sum of the degree measures in a triangle is 180, so:

R

4

T

m/RTP 1 m/RPT 1 m/4 5 180 30 1 125 1 m/4 5 180 m/4 5 25 RP bisects TRS, so mSRP m4 25. m/SRP 1 m/RPS 1 m/3 5 180 25 1 120 1 m/3 5 180 m/3 5 35 SP bisects RST, so m2 m3 35. m/1 1 m/2 1 m/STP 5 180 m/1 1 35 1 30 5 180 m/1 5 115


187


Class ______________

Chapter 9-5 Proving Triangles Congruent by Angle, Angle, Side

Date ______________


PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [6] g

1. In the figure given, FD ' DC and a b. E

G F a

2. Which of the following pairs of triangles can not be proved congruent by AAS? (1)

(3)

(2)

(4)

b D

C

Which additional information would permit you to prove that DFE DFG by the AAS theorem? ✔ (1) E G (2) EDF GDF (3) EF > GF (4) DE > DG

✔

3. Given: D lies on the angle bisector of ABC. Which of the following statements is true? h

(1) If DA ' BA, then DA BA. h

(2) If DC ' BC , then DC BC. h

h

✔ (3) If DA ' BA and DC ' BC , then DA DC. h

h

(4) If BA BC, then DA ' BA and DC ' BC .


188


Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 1 credit. No partial credit will be allowed. [6] 4. Complete the proof by filling in the missing reasons.

B

Given: AB > CB, ME ' AB, MF ' CB, and M is the midpoint of AC.

E

F

Prove: ME > MF A

M

C

Proof: Statements

Reasons

1. AB > CB

1. Given. 2. Isosceles triangle theorem.

2. A C 3. ME ' AB and MF ' CB 4. MEA and MFC are right angles.

3. Given. 4. Definition of perpendicular lines.

5. MEA MFC

5. Right angles are congruent.

6. M is the midpoint of AC. 7. AM > CM

6. Given. 7. Definition of midpoint.

8. AEM CFM

8. AAS (steps 2, 5, 7).

9. ME > MF

9. Corresponding parts of congruent triangles



Class ______________

Date ______________

PART III

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] C 5. Given: BC ' CD, BE ' ED, and 1 2 1 3 Prove: BCD BED D B A 2 4 Proof:

E

Statements

Reasons

1. BC ' CD and BE ' ED

1. Given.

2. BCE and BED are right angles.

2. Definition of perpendicular lines.

3. BCE BED


4. BD > BD

4. Reflexive property.

5. 1 2

5. Given.

6. 1 and 3 are supplements.

6. Definition of supplementary angles.

2 and 4 are supplements. 7. 3 4

7. If two angles are , then their supplements are .

8. BCD BED

8. AAS.

6. Given: AFEC, BF ' AC, DE ' AC, 1 2, AF > CE.

C

B 2

Prove: ADE CBF

E F 1

Proof:

A

Statements

Reasons

1. BF ' AC and DE ' AC

1. Given.

2. BFC and DEA are right angles.


3. BFC DEA


4. 1 2

4. Given.

5. AF > CE

5. Given.

6. AF 1 FE > FE 1 CE

6. Addition postulate.

7. AE 5 AF 1 FE FC 5 FE 1 EC

7. Partition postulate.

8. AE > FC

8. Substitution postulate.

9. ADE CBF

9. AAS.


D

189

190


Name __________________________________________________________

Class ______________

Chapter 9-6 The Converse of the Isosceles Triangle Theorem

Date ______________


PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. The measure of each base angle of an isosceles triangle is 15 degrees more than the measure of the vertex angle. What is the measure of the vertex angle? ✔ (1) 50°

(3) 30°

(2) 40°

(4) 25°

4. In PQR, mP x 37, mQ 3x 67, and mR = 14x 2 11. For the triangle to be isosceles, which of the following must be the value of x? (1) 64

✔ (3) 52

4 2011

(4) 89

(2)

2. The measure of vertex angle L of an isosceles triangle is three times the measure of each of the base angles, M and N. Which of the following statements is true? (1) MN 2LM

(3) LM LN MN

(2) LM LN MN

(4) 4LM 4LN ✔

3. In isosceles ABC, mB 40. Which statement can not be true? ✔ (1) AB BC and AC BC

5. If the degree measures of the three angles of a triangle are represented by x 25, 3x 15, and 4x 10, which of the following choices most completely describes the triangle? (1) scalene and right ✔ (2) isosceles and right (3) isosceles and acute (4) equilateral and equiangular 6. In each of the following, two angle measures of a triangle are given. Which of these could not be the angles of an isosceles triangle?

(2) mC mA (3) mA 100

(1) 70, 40

(4) AC BC

(2) 30, 120

(3) 80, 20 ✔ (4) 35, 65

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 7. a. Each of the congruent angles of an isosceles triangle measures 18 degrees less than 4 times the vertex angle. Find: (1) the measure of the vertex angle. Answer: 24° Solution: Let the measure of the vertex angle x. Then each base angle measures 4x 18. 4x 2 18 1 4x 2 18 1 x 5 180 9x 5 216 x 5 24



Class ______________

Date ______________

(2) the measure of the base angles. 4(24) 18 78° Answer

b. In ABC, A B. If AC 5x 7 and BC 2x 11, find: (1) the value of x. 5x 2 7 5 2x 1 11 3x 5 18 x 5 6 Answer (2) the lengths of AC and BC. Answer: AC BC 23 Solution: 5(6) 7 23 h

h

8. Given: QRA, QRS, RT bisects SRA, RT y QA

S R

Prove: a. QR > AR

1 2

T

b. QRA is isosceles.

Q

Proof: Statements

A

Reasons

h

a. 1. RT bisects SRA. 2. 1 2

1. Given. 2. Definition of angle bisector.

h

3. RT y QA

3. Given.

4. 1 Q

4. If two parallel lines are cut by a transversal, then the corresponding angles are .

5. 2 A

5. If two parallel lines are cut by a transversal, then the alternate interior angles formed are .

6. Q A


7. QR > AR

7. Converse of the isosceles triangle theorem.

b. 8. QRA is isosceles.



191

192


Name __________________________________________________________

Class ______________

Chapter 9-7 Proving Right Triangles Congruent by Hypotenuse-Leg

Date ______________


PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1.

B

A U

4. In right triangle ABC, angle bisectors AM, BR, and CU are drawn and intersect at point X. A

C M

U R

X

T

To prove ABC UTM by the hypotenuse-leg theorem, which additional corresponding parts must be congruent?

C

Which statement must be true?

(3) A and U

(1) AB and UT ✔ (2) AC and UM

(4) A and M

B

M

5.

(1) mRAX 45

(3) AC CM

(2) mRAX 45

✔ (4) AC CM

A

2. In the given figure, C is the midpoint of AE, BA ' AE, DE ' AE, and BC > DC. B

D

O

C

B

A

C

Which method of proof may be used to prove BAC DEC? (1) SAS (2) HA

D

E

✔ (3) HL (4) AAS

3. Two right triangles are not necessarily congruent if (1) the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle. (2) the hypotenuse and an acute angle of one triangle are congruent to the corresponding parts of the other triangle. ✔ (3) the corresponding acute angles of the triangles are congruent. (4) two legs of one triangle are congruent to two legs of the other triangle.

Which of the following is not sufficient to show that ABO DCO? (1) AB ' BC, DC ' BC, O is the midpoint of BC, and AB > CD. ✔ (2) DC ' BC, O is the midpoint of BC, and AB > CD. (3) AB ' BC, DC ' BC, and AB > CD. (4) AB ' BC, DC ' BC, and O is the midpoint of BC. 6. Which of the following statements is true? (1) HL can never be used to prove isosceles triangles congruent. (2) HL can never be used to prove acute triangles congruent. ✔ (3) HL is a special case of SSA. (4) HL is a special case of AAS. Copyright © Amsco School Publications, Inc.


Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 7. Given: In ABC, C is a right angle, DE ' BE, and BE > BC.

B

Prove: EBD CBD E

Proof:

D

A

Statements

Reasons

1. C is a right angle, DE ' BE, and BE > BC.

1. Given.

2. DEB is a right angle.


3. BD > BD


4. EBD CBD

4. HL.

8. Given: EM ' AD, BC ' AD, AM > DC, and AB > DE.

C

B

A M C

Prove: BC > EM E

D

Proof: Statements

Reasons

1. EM ' AD, BC ' AD

1. Given.

2. EMD and BCA are right angles.


3. AM > DC, AB > DE

3. Given.

4. AM 1 MC > MC 1 DC

4. Addition postulate.

5. AC 5 AM 1 MC, MD 5 MC 1 CD

5. Partition postulate.

6. AC > MD

6. Substitution postulate.

7. ABC DEM

7. HL.

8. BC > EM

8. Corresponding parts of congruent triangles are .


193

194


Name __________________________________________________________

Class ______________

Chapter 9-8 Interior and Exterior Angles of Polygons

Date ______________


PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the measures of the exterior angles at any two vertices of a triangle are equal, which of the following is false? (1) The triangle is equiangular.

4. What is the measure of one exterior angle of a regular pentagon? ✔ (1) 72° (2) 108°

(3) 360° (4) 540°

(2) The triangle is equilateral. ✔ (3) The triangle is concave. (4) The triangle is convex. 2. If the sum of the measures of the exterior angles of hexagon ABCDEF is 360°, which of the following must be true? (1) All exterior angles of ABCDEF are congruent. (2) All interior angles of ABCDEF are congruent.

5. The sum of the measures of the interior angles of a certain convex polygon is 720°. The sum of the measures of the interior angles of a second convex polygon that has two more sides than the first is (1) 720° (2) 900° ✔ (3) 1,080° (4) 1,440°

(3) ABCDEF is a regular polygon. ✔ (4) none of the above

6. The greatest measure that an exterior angle of any regular polygon can have is

3. If each exterior angle of a regular polygon contains 45°, how many sides does it have? (1) 6

✔ (3) 8

(2) 7

(4) 9

(1) 60° (2) 72°

(3) 90° ✔ (4) 120°

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 7. a. If the sum of the measures of eight interior angles of a nonagon is 1,180°, what is the measure of the ninth angle? Answer: 80° Solution: The sum of the measures of the interior angles of a nonagon is: 180(9 2) 1,260° The ninth angle measures 1,260 1,180 80°.



Class ______________

195

Date ______________

b. The measure of each exterior angle of a regular decagon is x 8. Find the value of x and the measure of each exterior angle. Answer: x 28°, exterior angle 36° Solution: A decagon has 10 sides. Each exterior angle measures 360 10 5 368. x 1 8 5 36 x 5 28 8. a. How many sides does a polygon have if the sum of the interior angles is four times the sum of the measures of its exterior angles? Answer: 10 sides Solution: The sum of the exterior angles 360°. Then the sum of the interior angles 4(360) 1,440°. 180(n 2 2) 5 1,440 n22 5 8 n 5 10 b. If an exterior angle of a regular polygon measures 3x and an interior angle measures 6x, how many sides does the polygon have? What is the name of the polygon? Answer: 6 sides, hexagon Solution: 180(n 2 2) 5 6x. n 720 3x, 360 n 5 3x or n 5 6x. 180(n 2 2) 5 720 n n

Since each interior angle measures 6x, Since each exterior measures

180(n 2 2) 5 720 n22 5 4 n56


196


Name __________________________________________________________

Class ______________


Date ______________

Chapter Review [40 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [16] 4. In ABC, if AB > BC, mA x 10 and mC 2x 20, what is the degree measure of B?

1. In the given figure, m ⊥ x and p m. m c

p

(1) 30 (2) 40

a b

✔ (3) 100

x

(4) 120 5. Each interior angle of a regular polygon has a measure of 140°. How many sides does the polygon have?

Which of the following must be false? (1) x ⊥ p (2) m p (3) ma mb ✔ (4) mc mb

(1) 8

(3) 10

✔ (2) 9

(4) 11

6. An exterior angle at the base of an isosceles triangle is always

2. If the coordinates of ABC are A(3, 4), B(4, 5), and C(4, 5), what is the slope of a line parallel to the line that passes through AB? ✔ (1) 297

(3) 17

(2) 279

(4) 7

(4) cannot be determined

m 105°

x°

(2) right ✔ (3) obtuse

3. In the given figure, r s and lines l and m are transversals. What is value of x? l

(1) acute

7. If the degree measures of the three angles of a triangle are represented by x 30, 4x 30, and 10x 30, which of the following choices most completely describes the triangle? (1) scalene and right

r

(2) isosceles and right ✔ (3) isosceles and acute

s 60°

(4) equilateral and equiangular 8. What is the measure of one interior angle of a regular octagon?

(1) 35

(3) 60

(1) 108°

✔ (3) 135°

✔ (2) 45

(4) 75

(2) 120°

(4) 144°


197


Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [4] g

g

9. In the given figure, parallel lines AB and CD are intersected by

F

g

transversal EF at points G and H, respectively. If mEGB x 40 and mGHD 4x 50, what is the value of x?

G A

B

Answer: 38 H

Solution:

D

C

x 1 40 1 4x 2 50 5 180

E

5x 2 10 5 180 5x 5 190 x 5 38

g

g

B

10. In the given figure, AB y DE, BE y AD, and m1 m2. Find the

x

measures of x and y. Answer: mx 50, my 65 Solution: m1 my my m2 mABE mDEF 130

A

(Alternate interior angles) (Substitution postulate) (Corresponding angles)

By the partition postulate: m/y 1 m/2 5 130

m/x 1 m/2 1 m/y 5 180

2m/y 5 130

m/x 1 130 5 180

m/y 5 65


m/x 5 50

2

y 1 C D

E 130°

198


Name __________________________________________________________

Class ______________

Date ______________

PART III

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] D

11. Line l is parallel to line m, and lines r and s are transversals. If DE DC, what is mABC?

l

C

Answer: mABC 100

A B

Solution: mDEC 180 140 40 mDCE mDEC 40 mACB mDCE 40 mBAC mDEC 40 mABC 180 mBAC mACB 180 40 40 100

E 140°

r

m

s

(Supplementary angles) (Base angles of an isosceles triangle) (Vertical angles) (Alternate interior angles) (Sum of the angle measures in a triangle)

12. Isosceles ABC with AB > BC, mA 4x 5, and mC 11x 23. a. Find mB. Answer: mB 138 Solution: 4x 1 5 5 11x 2 23 28 5 7x

m/B 5 180 2 21 2 21 5 138

x54 m/A 5 m/C 5 4(4) 1 5 5 21 b. True or False? AC AB. Justify your answer. Answer: False Explanation: If the measures of two angles of a triangle are unequal, then the lengths of the sides opposite these angles are unequal with the longer side opposite the larger angle. B C, so AC AB.



Class ______________

Date ______________

PART IV

Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] 13. The vertices of ABC are A(2, 0), B(6, 0), and C(8, 6). a. Write an equation of the line through point A parallel to BC. Answer: y 3x 6 Solution: 0 6 Slope of BC 5 68 2 26 5 2 5 3

y 5 3x 1 b 0 5 3(22) 1 b 65b Therefore, the equation of the line is y 3x 6. b. Write an equation of a line through point C parallel to AB. Answer: y 6 Solution: 020 Slope of AB 5 6 2 (22) 5 0

c. The lines whose equations were found in parts a and b intersect in point D. Find the coordinates of D. Answer: D(0, 6) Solution: 3x 1 6 5 6 3x 5 0 x50 y 5 3(0) 1 6 5 6


199

200


Name __________________________________________________________

Class ______________

14. Given: CH and AD are altitudes of ABC and 1 2.

B

Prove: ABC is isosceles.

H 1 A

Proof: Statements

Date ______________

D 2 C

Reasons

1. CH and AD are altitudes of ABC.

1. Given.

2. CH ' AB and AD ' BC

2. Definition of altitude.

3. CHA and ADC are right angles.


4. CHA ADC


5. 1 2

5. Given.

6. AC > AC


7. HAC DCA

7. AAS.

8. HAC DCA


9. AB > CB

9. Converse of the isosceles triangle theorem.

10. ABC is isosceles.

10. Definition of isosceles triangle.



Class ______________


201

Date ______________

Cumulative Review [40 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [16] 1. If the unequal sides of a triangle are 3, x, and 7, then what is the smallest integer value of x? (1) 3

✔ (3) 5

(2) 4

(4) 6

5. What is the slope of the line with the equation 3(x 1) 4y 0? (1) 0.75 ✔ (2) 0.75

2. Given: MAP with AQ a perpendicular bisector of MP

(3) 3 (4) 4

MQA is congruent to PQA by which of the following reasons?

6. If A → ~B and B are true, then which of the following is also true?

(1) SSS

✔ (1) ~A

(3) ASA

✔ (2) SAS

(4) HL

3. In the given figure, if point P is the same distance from the origin as point M, which of the following could be the coordinates of point P?

(2) ~B (3) A (4) cannot be determined 7. Under the transformation ry-axis + R908, the image of (5, 2) is

y M(a, b)

(1) (2, 5) (2) (2, 5) x

O

(4) (2, 5)

P

(1) (a, b) ✔ (2) (a, b)

✔ (3) (2, 5)

(3) (b, a) (4) (b, a)

4. Which of the following statements is true? (1) Every acute triangle is scalene. (2) Some right triangles are obtuse. ✔ (3) No scalene triangle is isosceles. (4) Some obtuse triangles are equilateral.


8. What are the coordinates of the midpoint of the line segment with endpoints at (a, b) and (g, h)? (1) (g a, h b) (2)

A

g2a h2b 2 , 2 B

A

g1a h1b 2 , 2 B

(3) (g a, h b) ✔ (4)

202


Name __________________________________________________________

Class ______________

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [4] 9. Find the sum of the measures of the interior angles of a 22-sided polygon. 180(22 2) 3,600° Answer

2

10. In the given figure, what is the average measure of the angles marked 1, 2, and 3? Answer: 60° Explanation: Each angle marked is a vertical angle with an angle of a triangle. The sum of the measures of the angles of a triangle is 180°, so the average is 180 3 5 608.

1

3

PART III

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [8] 11. Find the equation of the perpendicular bisector of the line segment with endpoints (3, 5) and (1, 3). Answer: y 5 212x 2 32 Solution: Midpoint of the given segment A 23 21 1, 25 21 3 B (1, 1) 3 2 (25)

Slope of given segment 1 2 (23) 5 84 5 2 Slope of perpendicular line 212

y 5 212x 1 b 21 5 212 (21) 1 b b 5 23 2 Therefore, the equation of the perpendicular bisector is y 5 212x 2 32.



Class ______________

Date ______________

12. The coordinates of DEF are D(2, 1), E(2, 7), and F(6, 0). Under a translation the image of D is D(1, 3). a. If the translation can be written as (x, y) → (x a, y b), what are the values of a and b? Answer: a 1, b 2 Solution: Since (2, 1) → (1, 3), the translation is defined by (x 1, y 2). b. Find the coordinates of E and F. Answer: E(3, 9), F(7, 2) Solution: T1,–2(2, 7) (3, 9)

T1,–2(6, 0) (7, 2)

PART IV

Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] 13. In ABC, AB 2x 3, BC 3x 1, and AC 4x 2. The perimeter of ABC is 36. a. Find the value of x. 2x 1 3 1 3x 2 1 1 4x 2 2 5 36 9x 5 36 x 5 4 Answer b. Find the length of each side. Answer: AB 11, BC 11, AC 14 Solution: AB 5 2x 1 3

BC 5 3x 2 1

AC 5 4x 2 2

5 2(4) 1 3

5 3(4) 2 1

5 4(4) 2 2

5 11

5 11

5 14

c. Name the largest angle or angles. Answer: B


203

204


Name __________________________________________________________

Class ______________

14. Given: ABC, BA > BC, AG > CR, DG ' AB, DR ' BC

Date ______________

B

Prove: DG > DR G A

Proof:

R D

Statements

Reasons

1. BA > BC, AG > CR

1. Given.

2. A C


3. DG ' AB, DR ' BC

3. Given.

4. DGA and DRC are right angles.


5. DGA DRC


6. DGA DRC

6. ASA.

7. DG > DR


C
