Download Chapter 9 Parallel LInes Answer Key

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 i 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 m/1 5 2x 1 40 and m/4 5 4x 2 10, find the value
of x so that AB y CD.
(1) 10
✔ (2) 25
If c 5 103, what is the value of a 1 b?
✔ (1) 154
(3) 180
(2) 177
(4) 206
5. In the given figure, if m1 i m2, what is the value of x?
(3) 45
40°
(4) 50
2. If m/2 5 x and m/3 5 3x 2 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 i m. Find the measure of /x.
6. In the given figure, if m1 ⊥ m2 and a 5 b, which of the
following statements is not true?
r
a
l
70°
35
m3
°
x
m
b
m2
m1
(1) 105
(2) 110
(3) 135
✔ (4) 145
Copyright © Amsco School Publications, Inc.
(1) m2 i 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 i d and lines a and b are transversals with line a
bisecting /QRS. If m/x 5 48, what is m/y?
a
Answer: m/y 5 24
Q
S
24°
24°
24° R
48°
x
c
24°
132°
d
y
E
b. In the given figure, m/BAC 5 100, m/CFG 5 130, AB y CD,
and CE y FG. Find m/DCE.
B
24°
G
D
Answer: m/DCE 5 50
50°
100° 80° 50°
A
C
130°
F
8. Given: BC > CA, m/6 5 m/3
C 3
1 2 4
Prove: l i 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. m/7 5 m/6
3. Definition of congruent angles.
4. m/6 5 m/3
4. Given.
5. m/7 5 m/3
5. Transitive property.
6. /7 > /3
6. Definition of congruent angles.
7. l i m
7. If two coplanar lines are cut by a transversal so that the
corresponding angles are >, then the two lines are i.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
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 i 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 i r and b 5 93, what is the
value of c 1 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°
Copyright © Amsco School Publications, Inc.
c
(1) 76
(3) 78
✔ (2) 77
(4) 79
182
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]
g
g
7. a. In the given figure, AB y CD, AE ' CB, m/BCD 5 2x, and
B
A
3x
m/BAE 5 3x. Find the value of x.
Answer: x 5 18°
E
2x
Solution:
Alternate interior angles formed by cutting parallel lines are congruent,
so m/ABE 5 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, m/BAD 5 33, and
m/BCD 5 58. Find m/DEF.
B
E
Answer: m/DEF 5 89
Solution:
Corresponding angles formed by cutting parallel lines are
congruent, so m/EDF 5 33 and m/EFD 5 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
g
2. AB y CD
1
H
2. If two coplanar lines are cut by a transversal so that the corresponding
angles are >, then the two lines are i.
3. /3 > /1
3. Given.
4. /2 > /1
4. Transitive property.
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 i.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
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 21 , then the slope of
g
CD is
(1) 22
(2) 212
✔ (3) 12
(4) 2
2. Find the equation of the line through (1, 22) and
parallel to the line with equation y 5 7x 2 2.
(1) y 5 217x 2 13
7
(2) y 5 217x 1 15
7
✔ (3) y 5 7x 2 9
(4) y 5 7x 1 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 5 1
(2) y 5 1
(3) x 5 4
✔ (4) y 5 4
Copyright © Amsco School Publications, Inc.
4. Which is the equation of a line passing through
(2, 25) and parallel to the line whose equation is
y 2 3x 5 2?
✔ (1) y 5 3x 2 11
(2) y 5 3x 2 5
(3) y 5 x 2 7
(4) y 5 3x 2 1
5. What is the slope of a line parallel to AB if
A(24, 23) and B(7, 21)?
(1) 11
2
2
✔ (2) 11
2
(3) 211
(4) 211
2
6. The lines x 1 2y 5 5 and 4x + ky 5 5 are parallel for
which value of k?
(1) 24
(2) 21
(3) 221
✔ (4) 8
184
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]
7. The coordinates of quadrilateral ABCD are A(25, 24), B(1, 22), C(2, 3), and D(24, 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 62 5 31
Slope of BC 5
321
1
2
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, 21), and
S(k, 23).
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
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9-4 The Sum of the Measures of the Angles of a Triangle
185
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]
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 i m, m/BAD 5 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 1 20, 5x 1 50, and 9x 2 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 i m, m/DAB 5 30,
m/ADB 5 125, and nABC 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°
Copyright © Amsco School Publications, Inc.
m
186
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]
7. In the given figure, m/D 5 45, m/A 5 35, and BC y DE. Find the degree
measures of the angles numbered 1 to 6.
D
Solution:
Since BC y DE, /D and /6 are congruent corresponding angles. Therefore,
m/6 5 m/D 5 45.
45°
B
4
6
Answer: m/1 5 80, m/2 5 100, m/3 5 80, m/4 5 135, m/5 5 100,
m/6 5 45
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 nRST, the bisectors of the angles meet at point P. If m/RTS 5 60, m/RPT 5 125,
and m/RPS 5 120, find the degree measures of the angles numbered 1 to 4.
32
Answer: m/1 5 115, m/2 5 35, m/3 5 35, m/4 5 25
Solution:
TP bisects /RTS, so m/RTP 5 m/STP 5 12 (60) 5 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 m/SRP 5 m/4 5 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 m/2 5 m/3 5 35.
m/1 1 m/2 1 m/STP 5 180
m/1 1 35 1 30 5 180
m/1 5 115
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187
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9-5 Proving Triangles Congruent by Angle, Angle, Side
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. [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 nDFE > nDFG 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 5 BA.
h
(2) If DC ' BC , then DC 5 BC.
h
h
✔ (3) If DA ' BA and DC ' BC , then DA 5 DC.
h
h
(4) If BA 5 BC, then DA ' BA and DC ' BC .
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188
Chapter 9 Parallel Lines
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. nAEM > nCFM
8. AAS (steps 2, 5, 7).
9. ME > MF
9. Corresponding parts of congruent triangles
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
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]
C
5. Given: BC ' CD, BE ' ED, and /1 > /2
1 3
Prove: nBCD > nBED
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
3. Right angles are congruent.
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. nBCD > nBED
8. AAS.
C
B
6. Given: AFEC, BF ' AC, DE ' AC, /1 > /2, AF > CE.
2
Prove: nADE > nCBF
E
F
1
Proof:
A
Statements
Reasons
1. BF ' AC and DE ' AC
1. Given.
2. /BFC and /DEA are right angles.
2. Definition of perpendicular lines.
3. /BFC > /DEA
3. Right angles are congruent.
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. nADE > nCBF
9. AAS.
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D
189
190
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9-6 The Converse of the Isosceles Triangle Theorem
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]
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°
(2) 40°
(3) 30°
4. In nPQR, m/P 5 x 1 37, m/Q 5 3x 2 67, and
m/R = 14x 2 11. For the triangle to be isosceles,
which of the following must be the value of x?
(1) 264
✔ (3) 52
4
2011
(4) 89
(2)
(4) 25°
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 5 2LM
(3) LM 1 LN 5 MN
(2) LM Þ LN Þ MN
(4) 4LM 5 4LN ✔
3. In isosceles nABC, m/B 5 40. Which statement can
not be true?
✔ (1) AB 5 BC and AC 5 BC
5. If the degree measures of the three angles of a
triangle are represented by x 1 25, 3x 2 15, and
4x 1 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) m/C . m/A
(3) m/A 5 100
(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 5 x.
Then each base angle measures 4x 218.
4x 2 18 1 4x 2 18 1 x 5 180
9x 5 216
x 5 24
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Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Date ______________
(2) the measure of the base angles.
4(24) 2 18 5 78° Answer
b. In nABC, /A > /B. If AC 5 5x 2 7 and BC 5 2x 1 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 5 BC 5 23
Solution:
5(6) 2 7 5 23
h
h
S
8. Given: nQRA, QRS, RT bisects /SRA, RT y QA
R
Prove: a. QR > AR
1
2
T
b. nQRA is isosceles.
Q
Proof:
Statements
h
a. 1. RT bisects /SRA.
2. /1 > /2
h
A
Reasons
1. Given.
2. Definition of angle bisector.
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
6. Transitive property.
7. QR > AR
7. Converse of the isosceles triangle theorem.
b. 8. nQRA is isosceles.
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8. Isosceles triangle theorem.
191
192
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9-7 Proving Right Triangles Congruent by Hypotenuse-Leg
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]
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 nABC > nUTM by the hypotenuse-leg
theorem, which additional corresponding parts must
be congruent?
(1) AB and UT
C
Which statement must be true?
(3) /A and /U
✔ (2) AC and UM
(4) /A and /M
B
M
5.
(1) m/RAX . 45
(3) AC 5 CM
(2) m/RAX 5 45
✔ (4) AC Þ 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
nBAC > nDEC?
(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
nABO > nDCO?
(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.
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. Given: In nABC, /C is a right angle, DE ' BE, and BE > BC.
Prove: nEBD > nCBD
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.
2. Definition of perpendicular lines.
3. BD > BD
3. Reflexive property.
4. nEBD > nCBD
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.
2. Definition of perpendicular lines.
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. nABC > nDEM
7. HL.
8. BC > EM
8. Corresponding parts of congruent triangles are >.
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194
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9-8 Interior and Exterior Angles of Polygons
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]
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 2) 5 1,260°
The ninth angle measures 1,260 2 1,180 5 80°.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
195
Date ______________
b. The measure of each exterior angle of a regular decagon is x 1 8. Find the value of x and the measure of each
exterior angle.
Answer: x 5 28°, exterior angle 5 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 5 360°.
Then the sum of the interior angles 5 4(360) 5 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
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196
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9 Parallel Lines
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 nABC, if AB > BC, m/A 5 x 1 10 and
m/C 5 2x 2 20, what is the degree measure of /B?
1. In the given figure, m ⊥ x and p i 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 i p
(3) m/a 5 m/b
(1) 8
(3) 10
✔ (2) 9
(4) 11
6. An exterior angle at the base of an isosceles triangle
is always
✔ (4) m/c . m/b
2. If the coordinates of nABC are A(23, 4), B(4, 25),
and C(4, 5), what is the slope of a line parallel to the
line that passes through AB?
✔ (1) 279
(3) 71
(2) 279
(4) 7
(4) cannot be determined
m
105°
x°
(2) right
✔ (3) obtuse
3. In the given figure, r i 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 1 30, 4x 1 30, and
10x 2 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°
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197
Chapter 9 Parallel Lines
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]
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 m/EGB 5 x 1 40 and
m/GHD 5 4x 2 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 m/1 5 m/2. Find the
x
measures of /x and /y.
Answer: m/x 5 50, m/y 5 65
Solution:
m/1 5 m/y
m/y 5 m/2
m/ABE 5 m/DEF 5 130
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
Copyright © Amsco School Publications, Inc.
m/x 5 50
2
y
1
C D
E
130°
198
Chapter 9 Parallel Lines
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 5 DC,
what is m/ABC?
l
C
Answer: m/ABC 5 100
A B
Solution:
r
m/DEC 5 180 2 140 5 40
m/DCE 5 m/DEC 5 40
m/ACB 5 m/DCE 5 40
m/BAC 5 m/DEC 5 40
m/ABC 5 180 2 m/BAC 2 m/ACB
5 180 2 40 2 40
5 100
E
140°
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 nABC with AB > BC, m/A 5 4x 1 5, and m/C 5 11x 2 23.
a. Find m/B.
Answer: m/B 5 138
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.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
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 nABC are A(22, 0), B(6, 0), and C(8, 6).
a. Write an equation of the line through point A parallel to BC.
Answer: y 5 3x 1 6
Solution:
6
20
Slope of BC 5 86 2
6 5 2 5 3
y 5 3x 1 b
0 5 3(22) 1 b
65b
Therefore, the equation of the line is y 5 3x 1 6.
b. Write an equation of a line through point C parallel to AB.
Answer: y 5 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
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200
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
14. Given: CH and AD are altitudes of nABC and /1 > /2.
B
Prove: nABC is isosceles.
H
1
A
Proof:
Statements
Date ______________
D
2
C
Reasons
1. CH and AD are altitudes of nABC.
1. Given.
2. CH ' AB and AD ' BC
2. Definition of altitude.
3. /CHA and /ADC are right angles.
3. Definition of perpendicular lines.
4. /CHA > /ADC
4. Right angles are congruent.
5. /1 > /2
5. Given.
6. AC > AC
6. Reflexive property.
7. nHAC > nDCA
7. AAS.
8. /HAC > /DCA
8. Corresponding parts of congruent triangles are >.
9. AB > CB
9. Converse of the isosceles triangle theorem.
10. nABC is isosceles.
10. Definition of isosceles triangle.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Chapter 9 Parallel Lines
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
23(x 1 1) 1 4y 5 0?
(1) 20.75
✔ (2) 0.75
2. Given: nMAP with AQ a perpendicular bisector
of MP
(3) 3
(4) 4
nMQA is congruent to nPQA 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, 22) is
y
M(a, b)
(1) (2, 5)
(2) (2, 25)
x
O
(4) (22, 25)
P
(1) (2a, b)
✔ (2) (a, 2b)
✔ (3) (22, 5)
(3) (2b, 2a)
(4) (2b, 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.
Copyright © Amsco School Publications, Inc.
8. What are the coordinates of the midpoint of the line
segment with endpoints at (a, b) and (g, h)?
(1) (g 2 a, h 2 b)
(2)
A
g2a h2b
2 , 2 B
(3) (g 1 a, h 1 b)
✔ (4)
A
g1a h1b
2 , 2 B
202
Chapter 9 Parallel Lines
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 2) 5 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 (23, 25) and (1, 3).
Answer: y 5 212x 2 32
Solution:
Midpoint of the given segment 5 A 2321 1, 2521 3 B 5 (21, 21)
3 2 (25)
Slope of given segment 5 1 2 (23) 5 84 5 2
Slope of perpendicular line 5 212
y 5 221x 1 b
21 5 212 (21) 1 b
b 5 23
2
Therefore, the equation of the perpendicular bisector is y 5 221x 2 23.
Copyright © Amsco School Publications, Inc.
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Date ______________
12. The coordinates of nDEF are D(22, 21), E(2, 27), and F(6, 0). Under a translation the image of D is
D9(21, 23).
a. If the translation can be written as (x, y) → (x 1 a, y 1 b), what are the values of a and b?
Answer: a 5 1, b 5 22
Solution:
Since (22, 21) → (21, 23), the translation is defined by (x 1 1, y 1 22).
b. Find the coordinates of E9 and F9.
Answer: E9(3, 29), F9(7, 22)
Solution:
T1,–2(2, 27) 5 (3, 29)
T1,–2(6, 0) 5 (7, 22)
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 nABC, AB 5 2x 1 3, BC 5 3x 2 1, and AC 5 4x 2 2. The perimeter of nABC 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 5 11, BC 5 11, AC 5 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
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204
Chapter 9 Parallel Lines
Name __________________________________________________________
Class ______________
Date ______________
B
14. Given: nABC, BA > BC, AG > CR, DG ' AB, DR ' BC
Prove: DG > DR
G
A
Proof:
R
D
Statements
Reasons
1. BA > BC, AG > CR
1. Given.
2. /A > /C
2. Isosceles triangle theorem.
3. DG ' AB, DR ' BC
3. Given.
4. /DGA and /DRC are right angles.
4. Definition of perpendicular lines.
5. /DGA > /DRC
5. Right angles are congruent.
6. nDGA > nDRC
6. ASA.
7. DG > DR
7. Corresponding parts of congruent triangles are >.
C
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