Name: ________________________ Class: ___________________ Date: __________
Ch 3 Review ____
1. In the diagram, c Ä d . Identify three numbered angles that have a measure of 124°.
a. ____
b.
∠2, ∠4, and ∠1
c.
∠3, ∠7, and ∠4
d.
∠2, ∠3, and ∠7
c.
–15
d.
195
c.
36
d.
20
2. In the diagram, a Ä b . Find the value of x.
a. ____
∠2, ∠7, and ∠1
125
b.
55
3. In the diagram, b Ä c. Find the value of y.
a.
70
b.
54
1
ID: A
← →
____
4. In the diagram, BC bisects ∠ACD and j Ä k . Find the value of w.
a. ____
b.
–22
c.
58
d.
–13
c.
42
d.
40
c.
15
d.
55
5. Find the value of x that makes j Ä k .
a. ____
67
44
b.
46
6. Find the value of x that makes a Ä b .
a.
130
b.
50
2
____
7. Given a Ä b. Find the value of z that makes j Ä k .
a. ____
b.
z = 11.25
c.
z = 17.25
d.
z = –48
c.
10 units
d.
about 2.8 units
jÄk
d.
nÄl
8. Find the distance from point P to RQ .
a. ____
z = 228
about 3.2 units
b.
4 units
9. Which two lines must be parallel in the diagram?
a.
mÄ n
b.
lÄm
c.
3
____ 10. The diagram represents the streets of the downtown area of a city. Determine which lines, if any, must be parallel in the diagram.
a.
uÄt
b.
aÄb
c.
sÄu
d.
none
____ 11. Write an equation of the line passing through the point ( 6, –2) that is parallel to the line y = 4x − 11 . 1 a. y = − x − 2 c. y = 4x − 2 4 1 b. y = − x − 26 d. y = 4x − 26 4 ____ 12. Write the point slope equation of the line passing through the point (–2, –5) and parallel to the line y = 3x − 4. 1 a. (y + 5) = 3(x + 2) c. (y + 5) = (x + 2) 3 1 b. (y - 5) = 3(x - 2) d. (y - 5) = - (x - 2) 3 ____ 13. Write an equation of the line passing through the point (8,2) that is parallel to the line 6x − 9y = −7. 2 10 2 c. y = x + 6 a. y = x − 3 3 3 2 2 7 b. y = x + 30 d. y = x + 3 3 9 ____ 14. Write an equation of the line passing through the point ( 8, 7) that is perpendicular to the line 7 y + 7 = − (x + 8 ) . 17 17 7 x+7 a. y = − x + 7 c. y = 7 17 7 63 17 87 x− b. y = − x + d. y = 17 17 7 7
4
____ 15. Write an equation of the line passing through the point ( –1, 5) that is perpendicular to the line y = − a. b.
13 x+5 9 13 58 y= x+ 9 9
c.
y=−
d.
9 x + 8. 13
9 74 x+ 13 13 9 y= x+5 13 y=−
____ 16. Write the equation of the line through (2, 8) and parallel to y = -6 a. y = 8 c. x = 8 b. y = -6 d. x = 2 ____ 17. Write an equation of the line through the point (−3,−8) and perpendicular to the line y = −7 . 1 a. x = −3 c. y = 7 b. x = −8 d. y = −8 ____ 18. The diagram shows a right rectangular prism with two parallel segments intersected by a transversal on the front face. Think of each segment in the diagram as part of a line. Which statements are true?
a.
∠6 and ∠1 are alternate exterior angles. ← →
b. c.
← →
d.
← →
AB is perpendicular to CF . ∠6 and ∠1 are alternate interior angles. ← →
AB is skew to CF .
5
____ 19. In the diagram, a Ä b, a Ä c, a Ä d, b Ä c, b Ä d, and c Ä d. Which statements about the diagram are true?
a. b. c. d.
The value of x is 41. The value of y is 20. ∠2 ≅ ∠3 by the Vertical Angles Congruence Theorem. ∠5 ≅ ∠6 by the Corresponding Angles Theorem.
____ 20. Which statements are enough to conclude that m Ä n ?
a. b.
∠6 ≅ ∠3 ∠1 and ∠7 are supplementary.
c. d.
Match the description below with its equation. 1 11 a. y = − x + d. 4 2 b. y = 9x –2 e. 1 c. y = x + 88 f. 4
∠1 ≅ ∠9 ∠1 ≅ ∠12
y = 5x + 29
y = 9x + 7 y = 5x + 7
1 ____ 21. line passing through the point (1, 7) that is perpendicular to the line y = − x + 6 9
6
____ 22. perpendicular bisector of PQ with endpoints P(4, –4) and Q(8, 12) ____ 23. line passing through the point (–4, 9) that is parallel to the line y = 5x + 4 ____ 24. Classify the pair of numbered angles.
a. b.
corresponding alternate exterior
c. d.
alternate interior consecutive interior
c. d.
alternate interior consecutive interior
c. d.
alternate interior consecutive interior
c. d.
alternate interior consecutive interior
____ 25. Classify the pair of numbered angles.
a. b.
corresponding alternate exterior
____ 26. Classify the pair of numbered angles.
a. b.
corresponding alternate exterior
____ 27. Classify the pair of numbered angles.
a. b.
corresponding alternate exterior
7
____ 28. Classify the pair of numbered angles.
a. b.
corresponding alternate exterior
c. d.
alternate interior consecutive interior
The given line markings show how the roads in a town are related to one another. 29. Name a pair of parallel lines.
30. Name a pair of parallel lines.
8
→
← →
31. Is CA ⊥ BF ? Explain.
← →
← →
32. Is AB ⊥ EF ? Explain.
33. A rope for zip lining is tied between two parallel trees. The rope is taut, forming a straight line. What is m∠2? How do you know?
9
34. In the diagram of the parking lot, m∠1 = 60°. What is m∠2? How do you know?
35. A sailor sees a lighthouse at a 55° angle of elevation from the horizontal. In the diagram, m∠2 = 55°. What is m∠1? How do you know?
36. A helicopter pilot sees an island at a 70 ° angle of depression from the horizontal. In the diagram, m∠1 = 70°. What is m∠2? How do you know?
10
37. Each railroad tie in the track is parallel to the tie immediately below it. Explain why the top tie is parallel to the bottom tie.
Which lines, if any, can be proved parallel given the following diagram? 38.
True or False: 39. If two lines are intersected by a transversal and alternate interior angles are equal in measure, then the lines are parallel. 40. If two lines are intersected by a transversal and corresponding angles are supplementary, then the lines are parallel. ____ 41. What is the slope of the line that passes through points A (-2,-3) and B (5,3)? 6 6 a. c. − 7 7 7 b. d. 0 6 ____ 42. Which best describes the relationship between the line that passes through (–1, –8) and (4, –4) and the line that passes through (2, –9) and (7, –5)? a. parallel b. perpendicular c. same line d. neither perpendicular nor parallel 43. Decide whether Line 1 and Line 2 are parallel, perpendicular, or neither. Line 1 passes through (10, 7) and (13, 9) Line 2 passes through (–4, 3) and (–1, 5)
11
____ 44. Which best describes the relationship between Line 1 and Line 2? Line 1 passes through ÊÁË −3, 6 ˆ˜¯ and ÊÁË −7, 11 ˆ˜¯ Line 2 passes through ÁÊË 1, 8 ˜ˆ¯ and ÁÊË −4, 4 ˜ˆ¯ a. perpendicular b. They are the same line. c. parallel d. neither perpendicular nor parallel 45. What is the slope of a line parallel to the line 3x − 2y = 8?
1 1 46. Are the lines with the equations y = − x + 2 and y = − x − 2 parallel, perpendicular, or skew? Explain 3 3 your answer. ____ 47. Which line is parallel to y = a. b.
2 x − 7? 3
3 y = − x+7 2 2 y = x+1 3
c. d.
3 x+2 2 2 y = − x−7 3
y=
48. Write an equation for the line passing through the point ÊÁË −2, 4 ˆ˜¯ that has a slope of 3. 49. Write the slope-intercept form of the equation of the line passing through the point (–2, –5) and parallel to the line y = 3x − 4. ____ 50. Find the slope-intercept form of the the line passing through the point ÊÁË −3, − 6 ˆ˜¯ and parallel to the line y = 5x + 4. 1 33 a. y = −5x − 9 c. y = − x − 5 5 b. y = 5x + 9 d. y = 5x + 27 ____ 51. Which line is perpendicular to y = −4x + 2 ? a.
y = 4x + 2
c.
b.
y = −4x − 2
d.
12
1 x+1 4 1 y = − x+1 4 y=
____ 52. In the figure, l Ä n and r is a transversal. Which of the following is not necessarily true?
a. b.
∠8 ≅ ∠2 ∠2 ≅ ∠6
c. d.
∠5 ≅ ∠3 ∠7 ≅ ∠4
Which lines, if any, can be proved parallel given the following diagram? For each conclusion, provide the justification. 53.
13
54. Use the figure and the given information to determine which lines must be parallel. Given: ∠1 ≅ ∠3
55. Use the figure and the given information to determine which lines must be parallel. Given: ∠1 ≅ ∠3
14
____ 56. Identify the transversal and classify the angle pair ∠11 and ∠7.
a. b. c. d.
The transversal is line The transversal is line The transversal is line The transversal is line
l. The angles are corresponding angles. l. The angles are alternate interior angles. n. The angles are alternate exterior angles. m. The angles are corresponding angles.
____ 57. Violin strings are parallel. Viewed from above, a violin bow in two different positions forms two transversals to the violin strings. Find x and y in the diagram.
a. b.
c. d.
x = 10, y = 20 x = 20, y = 20
x = 0, y = 60 x = −10, y = 140
____ 58. Find the value of x for which l is parallel to m. The diagram is not to scale.
a.
35
b.
145
c.
15
70
d.
105
____ 59. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a. b.
m∠1 = 95° m∠1 = 80°
c. d.
m∠1 = 85° m∠1 = 75°
____ 60. Write the equation of the line with slope 2 through the point (4, 7) in point-slope form. a. y = 2x − 1 c. y − 4 = 2(x − 7) b. y = 2x + 7 d. y − 7 = 2(x − 4) ____ 61. Write the equation of the perpendicular bisector of the segment with endpoints (0, 5) and (8, 1) (in point slope form) a. (y - 6) = 2(x - 0) c. (y - 4) = 2(x - 3) b. (y - 3) = 2(x - 4) d. (y - 1) = 2(x - 8) 62. Given: m Ä n and ∠1 is congruent to ∠3. Prove r Ä s using a 2 column proof.
.
16
63. Given: r Ä s and ∠1 is supplementary to ∠3. Prove m Ä n using a two column proof.
. 64. Given: m Ä n and ∠2 is congruent to ∠3. Prove j Ä k using a 2 column proof.
.
17
65. Given: p Ä q and ∠2 is congruent to ∠3. Prove r Ä s using a 2 column proof.
. 66. Given: r Ä s and ∠1 is congruent to ∠3. Prove m Ä n using a two column proof.
18
ID: A
Ch 3 Review Answer Section 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
D B C D C B D D A A D A A D B A A C, D A, C A, B B A D B A C D A ← →
← →
29. AF Ä BE ← →
← →
30. AD Ä BF →
← →
→
← →
31. no; CA is not perpendicular to BF , because DM ⊥ BF . By the Perpendicular Postulate, there is exactly one ← →
line through M that is perpendicular to BF . ← →
← →
← →
← →
32. no; AB is not perpendicular to EF , because EF ⊥CD . By the Perpendicular Postulate, there is exactly one ← →
33. 34. 35. 36. 37.
line through D that is perpendicular to EF . m∠2 = 108°; Consecutive Interior Angles Theorem m∠2 = 60°; Alternate Exterior Angles Theorem m∠1 = 55°; Alternate Interior Angles Theorem m∠2 = 70°; Alternate Interior Angles Theorem By The Transitive Property of Parallel Lines, the top tie is parallel to the bottom tie. 1
ID: A 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.
63.
64.
65.
66.
No lines can be proved parallel from the given information. True False A A parallel A 3 2 parallel; Slopes are equal and y-intercepts are different B y = 3x + 10 y = 3x + 1 B C D b Ä c, Consecutive Interior Angles Converse l Äm nÄp A A A C D C It is given that ∠1 is congruent to ∠3. It is also given that m Ä n, so by the Alternate Interior Angles Theorem, ∠2 ≅ ∠3. Then ∠1 ≅ ∠2 by the Transitive Property of Congruence. So, by the Alternate Exterior Angles Converse, r Ä s. It is given that ∠1 is supplementary to ∠3, so m∠1 + m∠3 = 180° by the definition of supplementary angles. It is also given that r Ä s, so by the Corresponding Angles Theorem, ∠1 ≅ ∠2. By the definition of congruent angles, m∠1 = m∠2. By the Substitution Property of Equality, m∠2 + m∠3 = 180°. So, ∠2 and ∠3 are supplementary by the definition of supplementary angles. Then, by the Consecutive Interior Angles Converse, m Ä n . It is given that ∠2 is congruent to ∠3. It is also given that m Ä n, so by the Alternate Interior Angles Theorem, ∠1 ≅ ∠2. Then ∠1 ≅ ∠3 by the Transitive Property of Congruence. So, by the Alternate Interior Angles Converse, j Ä k . It is given that ∠2 is congruent to ∠3. It is also given that p Ä q, so by the Corresponding Angles Theorem, ∠1 ≅ ∠2. Then ∠1 ≅ ∠3 by the Transitive Property of Congruence. So, by the Alternate Exterior Angles Converse, r Ä s. It is given that ∠1 is congruent to ∠3. It is also given that r Ä s, so by the Alternate Exterior Angles Theorem, ∠2 ≅ ∠3. Then ∠1 ≅ ∠2 by the Transitive Property of Congruence. So, by the Corresponding Angles Converse, m Ä n .
2
