# Parallel Lines Chapter Problems -...

### Transcript of Parallel Lines Chapter Problems -...

Geometry: Parallel Lines ~1~ NJCTL.org

Parallel Lines Chapter Problems Lines: Intersecting, parallel & skew Class Work – Use image 1

1. Name all segments parallel to 𝐺𝐻̅̅ ̅̅ :

2. Name all segments skew to 𝐺𝐻̅̅ ̅̅ :

3. Name all segments intersecting with 𝐺𝐻̅̅ ̅̅ :

4. Are segments 𝐺𝐻̅̅ ̅̅ and 𝐵𝐴̅̅ ̅̅ coplanar? Explain your answer.

5. Are segments 𝐺𝐻̅̅ ̅̅ and 𝐵𝐹̅̅ ̅̅ coplanar? Explain your answer.

Is each statement true always, sometimes, or never? 6. Two intersecting lines are skew. 7. Two parallel lines are coplanar. 8. Two lines in the same plane are parallel. 9. Two lines that do not intersect are parallel. 10. Two skew lines are coplanar Lines: Intersecting, parallel & skew Homework -Use Image 1

11. Name all segments parallel to 𝐹𝐸̅̅ ̅̅ :

12. Name all segments skew to 𝐹𝐸̅̅ ̅̅ :

13. Name all segments intersecting with 𝐹𝐸̅̅ ̅̅ :

14. Are segments 𝐹𝐸̅̅ ̅̅ and 𝐶𝐷̅̅ ̅̅ coplanar? Explain your answer.

15. Are segments 𝐹𝐸̅̅ ̅̅ and 𝐻𝐷̅̅ ̅̅ coplanar? Explain your answer. State whether the following statements are always, sometimes, or never true: 16. Two coplanar lines are skew. 17. Two intersecting lines are in the same plane. 18. Two lines in the same plane are parallel. Lines & Transversals Classify each pair of angles as alternate interior, alternate exterior, same-side interior, same-side exterior, corresponding angles, or none of these. 19. ∠11 and ∠16 are

20. ∠12 and ∠2 are

21. ∠14 and ∠8 are 22. ∠6 and ∠16 are

23. ∠7 and ∠14 are

24. ∠3 and ∠16 are

Image 1

Geometry: Parallel Lines ~2~ NJCTL.org

Classify each pair of angles as alternate interior, alternate exterior, same-side interior, same-side exterior, corresponding angles, or none of these.

25. ∠7 and ∠12 26. ∠3 and ∠6 27. ∠6 and ∠11

28. ∠7 and ∠11 29. ∠4 and ∠10 30. ∠14 and ∠16 31. ∠2 and ∠3

32. ∠2 and ∠10 Parallel Lines & Proofs Classwork Match each expression/equation with the property used to make the conclusion. 33. AB = AB

34. If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C.

35. If x + y = 9 and y = 5, then x + 5 = 9.

36. If DE = FG, then FG = DE. a) Substitution Property of Equality b) Transitive Property of Equality c) Reflexive Property of Equality d) Symmetric Property of Equality

PARCC type question: 37. Alternate Exterior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Given: line m || line k

Prove: ∠2 ≅ ∠8

Statements Reasons

1. line m || line k 1.

2. ∠2 ≅ ∠6 2.

3. ∠6 ≅ ∠8 3.

4. ∠2 ≅ ∠8 4.

Reasons Bank a) Transitive Property of Congruence b) If 2 parallel lines are cut by a

transversal, then the corresponding angles are congruent.

c) Vertical Angles are congruent. d) Given

Geometry: Parallel Lines ~3~ NJCTL.org

PARCC type question: 38. Same-Side Interior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Some reasons may be used more than once. Given: line m || line k

Prove: ∠5 & ∠4 are supplementary

Parallel Lines & Proofs Homework For #39-42 match the description on the left to the name of the property on the right.

39. ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. a) Substitution Property of Equality 40. If bc = 77 and b = 11, then 11c = 77. b) Transitive Property of Congruence

41. If ∠P ≅ ∠M, then ∠M ≅ ∠P. c) Reflexive Property of Equality 42. QR = QR d) Symmetric Property of Congruence

Statements Reasons

1. line m || line k 1.

2. ∠1 ≅ ∠5 2.

3. m∠1 = m∠5 3.

4. ∠1 & ∠4 are supplementary 4.

5. m∠1 + m∠4 = 180 5.

6. m∠5 + m∠4 = 180 6.

7. ∠5 & ∠4 are supplementary 7.

Reasons Bank

a) Angles that form a linear pair are supplementary.

b) Substitution Property of Equality c) Definition of supplementary angles d) If 2 parallel lines are cut by a

transversal, then the corresponding angles are congruent.

e) Definition of congruent angles f) Given

Geometry: Parallel Lines ~4~ NJCTL.org

PARCC type question: 43. Alternate Interior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Given: line m || line k

Prove: ∠3 ≅ ∠5

Statements Reasons

1. line m || line k 1.

2. ∠3 ≅ ∠7 2.

3. ∠7 ≅ ∠5 3.

4. ∠3 ≅ ∠5 4.

Reasons Bank

a) Vertical Angles are congruent. b) Given c) Transitive Property of Congruence d) If 2 parallel lines are cut by a transversal,

then the corresponding angles are congruent.

Geometry: Parallel Lines ~5~ NJCTL.org

PARCC type question: 44. Same-Side Exterior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Some reasons may be used more than once. Given: line m || line k

Prove: ∠1 & ∠8 are supplementary

Properties of Parallel Lines Classwork Use the given diagram to answer problems #33-41.

If m∠9 = 54°, then find the measure the following angles:

45. m∠1=

46. m∠2=

47. m∠4=

48. m∠5=

49. m∠15=

Statements Reasons

1. line m || line k 1.

2. ∠1 ≅ ∠5 2.

3. m∠1 = m∠5 3.

4. ∠5 & ∠8 are supplementary

4.

5. m∠5 + m∠8 = 180 5.

6. m∠1 + m∠8 = 180 6.

7. ∠1 & ∠8 are supplementary

7.

Reasons Bank

a) Definition of supplementary angles b) If 2 parallel lines are cut by a transversal,

then the corresponding angles are congruent.

c) Given d) Definition of congruent angles e) Angles that form a linear pair are

supplementary. f) Substitution Property of Equality

Geometry: Parallel Lines ~6~ NJCTL.org

If m∠2 = (12x-54)° and m∠10 = (7x+26)°, then find the measure the following angles: 50.m∠6= 51. m∠11=

52. m∠9= 53. m∠16= Find the values of the unknown variables in each figure. (# 54-58) 54. 55. 56.

57. 58.

Geometry: Parallel Lines ~7~ NJCTL.org

Find measure of the following angles:

59. m∠1= 60. m∠2= 61. m∠3=

62. m∠4= 63. m∠5= State which segments (if any) are parallel. 64. 65. 66. Solve for the unknowns 67. 68.

Geometry: Parallel Lines ~8~ NJCTL.org

Properties of Parallel Lines Homework

If m∠9 = 62°, then find the measure the following angles: 69. m∠1= 70. m∠2=

71. m∠4= 72. m∠5= 73. m∠15=

If m ∠2 = (14x-24)° and m ∠10 = (6x+72)°, then find the measure the following angles:

74. m∠6= 75. m∠11= 76. m∠9= 77. m∠16= Find the values of the unknown variables in each figure. (#78-82) 78. 79. 80. 81. 82.

Geometry: Parallel Lines ~9~ NJCTL.org

Find measure of the following angles:

83. m∠1= 84. m∠2= 85. m∠3=

86. m∠4= 87. m∠5= State which segments (if any) are parallel. 88.

90. 89. 91. 92.

124°

124°

D C

BA

Geometry: Parallel Lines ~10~ NJCTL.org

Constructing Parallel Lines Class Work 93. Construct a line m that is parallel to line l that passes thru point C using the stated method. Corresponding Angles 94. Error Analysis: A person was constructing the line n thru point D such that it

was parallel to line l using the alternate interior angles method. Using their markings, state their mistake.

95. Use paper- folding techniques to construct parallel lines.

Geometry: Parallel Lines ~11~ NJCTL.org

Constructing Parallel Lines Homework 96. Error Analysis: A person was constructing the line n thru point D such that it

was parallel to line l using the alternate exterior angles method. Using their markings, state their mistake.

97. Construct parallel lines using a straightedge and compass using alternate interior angles. 98. Construct parallel lines using a straightedge and compass using alternate exterior angles.

Geometry: Parallel Lines ~12~ NJCTL.org

PARCC type question: 99. The figure shows line j, points C and B are on line j, and point A is not on line j. Also shown is line AB.

Part A:

Consider the partial construction of a line parallel to j through point A. What would be the final step in the construction?

a) Draw a line through points B and F b) Draw a line through points C and F c) Draw a line through points A and F d) Draw a line through points A and G

Part B: Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction?

a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary. d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

jBC

A

jG

F

BC

A

Geometry: Parallel Lines ~13~ NJCTL.org

PARCC type question: 100. The figure shows line p; points H, K, and M are on line p, and point J is not on line p. Also shown is line JK.

Part A:

Consider the partial construction of a line parallel to p through point J. What would be the final step in the construction?

a) Draw a line through points K and N b) Draw a line through points J and N c) Draw a line through points H and N d) Draw a line through points M and M

Part B: Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction?

a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent. b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary. d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

pMKH

J

p

N

MKH

J

Geometry: Parallel Lines ~14~ NJCTL.org

Parallel Lines Review Multiple Choice

1. Name the segment parallel to 𝐺𝐻̅̅ ̅̅ and skew to 𝐸𝐴̅̅ ̅̅ .

a. 𝐹𝐵̅̅ ̅̅

b. 𝐷𝐴̅̅ ̅̅

c. 𝐽�̅�

d. 𝐻𝐷̅̅ ̅̅

2. Name the segment parallel to 𝐵𝐶̅̅ ̅̅ and skew to 𝐸𝐼.̅̅ ̅̅

a. 𝐹𝐵̅̅ ̅̅

b. 𝐷𝐴̅̅ ̅̅

c. 𝐽�̅�

d. 𝐻𝐷̅̅ ̅̅

3. Determine if the statement is always, sometimes, or never true:

Two skew lines are coplanar. a. Always b. Sometimes

c. Never

4. Determine if the statement is always, sometimes, or never true: Two intersecting lines are coplanar a. Always b. Sometimes

c. Never 5. Determine if the statement is always, sometimes, or never true:

Two lines that do not intersect are skew. a. Always b. Sometimes c. Never

6. Determine the relationship between ∠1 & ∠10. a. Alternate Interior b. Same-side Interior c. Corresponding Angles d. None of these

7. Determine the relationship between ∠5 & ∠15. a. Alternate Exterior b. Alternate Interior

c. Same-side Interior d. None of these

Geometry: Parallel Lines ~15~ NJCTL.org

8. Given in the diagram to the right, m∠2=3x-10 and m∠15=2x+30 , what is m∠12? a. 32o b. 40o c. 86o d. 110o

9. Given in the diagram to the right, m∠5= (7x+2)°and m∠11=(5x+14)°, what is

m∠14? a. 6° b. 44° c. 46° d. 136°

In 10-11, use the diagram at the right.

10. Given ∠2 ≅ ∠6, what justifies k || m. a. Converse Alternate Interior Angles Theorem b. Converse Alternate Exterior Angles Theorem c. Converse Corresponding Angles Theorem d. there is not enough info to state parallel

11. Given n || p , what justifies ∠1 ≅ ∠12 a. Alternate Interior Angles Theorem b. Alternate Exterior Angles Theorem c. Corresponding Angles Theorem d. there is not enough info to make this statement

Extended Constructed Response 1. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once.

Given: ∠1 ≅ ∠3; 𝑀𝑁̅̅ ̅̅ ̅ || 𝑃𝑄̅̅ ̅̅

Prove: ∠2≅∠3

Statements Reasons

1. ∠1 ≅ ∠3 1.

2. 𝑀𝑁̅̅ ̅̅ ̅ || 𝑃𝑄̅̅ ̅̅ 2.

3. ∠1 ≅ ∠2 3.

4. ∠2≅∠3 4.

3 2

1

M N

QP

Reasons Bank

a) Transitive Property of Congruence

b) If 2 parallel lines are cut by a

transversal, then the alternate interior angles

are congruent.

c) Given

Geometry: Parallel Lines ~16~ NJCTL.org

2. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once. Given: n || p, k || m

Prove: ∠2 & ∠13 are supplementary

Statements Reasons

1. n || p, k || m 1.

2. ∠2 ≅ ∠12 2.

3. ∠12 ≅ ∠14 3.

4. ∠2 ≅ ∠14 4.

5. m∠2 = m∠14 5.

6. m∠13 & m∠14 are supplementary

6.

7. m∠13 + m∠14 = 180° 7.

8. m∠13 + m∠2 = 180° 8.

9. ∠2 &∠13 are supplementary 9.

3. Using a compass and straightedge, construct parallel lines. You can use any method of your choice.

Reasons Bank

a) Transitive Property of Congruence b) Definition of supplementary angles c) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. d) Definition of Congruent Angles e) Given f) If 2 parallel lines are cut by a transversal, then the alternate exterior angles are congruent. g) Angles that form a linear pair are supplementary h) Substitution Property of Equality

Geometry: Parallel Lines ~17~ NJCTL.org

Answers

1. Segments 𝐽�̅�,𝐹𝐸̅̅ ̅̅ , 𝐵𝐴̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅

Segments 𝐽𝐹̅̅ ̅,𝐼𝐸̅̅ ̅, 𝐹𝐵̅̅ ̅̅ , 𝐸𝐴̅̅ ̅̅

2. Segments 𝐺𝐽̅̅ ̅, 𝐺𝐹̅̅ ̅̅ , 𝐺𝐶̅̅ ̅̅ ,

𝐻𝐼̅̅̅̅ , 𝐻𝐸̅̅ ̅̅ , 𝐻𝐷̅̅ ̅̅

3. Yes, because these segments

are parallel

4. No, these lines are skew, so

they are not coplanar.

5. Never

6. Always

7. Sometimes

8. Sometimes

9. Never

10. Segments 𝐽�̅�, 𝐺𝐻̅̅ ̅̅ , 𝐵𝐴̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅

11. Segments 𝐻𝐼̅̅̅̅ , 𝐺𝐽̅̅ ̅, 𝐶𝐺̅̅ ̅̅ , 𝐷𝐻̅̅ ̅̅

12. Segments 𝐹𝐺̅̅ ̅̅ , 𝐹𝐵̅̅ ̅̅ , 𝐹𝐽̅̅ ̅, 𝐸𝐴̅̅ ̅̅ , 𝐹𝐵̅̅ ̅̅ ,

𝐸𝐻̅̅ ̅̅ , 𝐸𝐼̅̅ ̅

13. Yes, because they are parallel

14. No, these lines are skew, so

they are not coplanar

15. Never

16. Always

17. Sometimes

18. Same side interior

19. None of these

20. Alternate interior

21. Corresponding

22. Same-side interior

23. None of these

24. Corresponding

25. Same-side

26. Alternate interior

27. Corresponding

28. Corresponding

29. Same-side interior

30. None of these

31. None of these

32. c. Reflexive Property of

Equality

33. b. Transitive Property of

Equality

34. a. Substitution Property of

Equality

35. d. Symmetric Property of

Equality

36. Proof reasons should be:

Statements Reasons

1. line m || line k 1. d.

2. ∠2 ≅ ∠6 2. b.

3. ∠6 ≅ ∠8 3. c.

4. ∠2 ≅ ∠8 4. a.

37. Proof reasons should be:

Statements Reasons

1. line m || line k 1. f.

2. ∠1 ≅ ∠5 2. d.

3. m∠1 = m∠5 3. e.

4. ∠1 & ∠4 are supplementary

4. a.

5. m∠1 + m∠4 = 180°

5. c.

6. m∠5 + m∠4 = 180°

6. b.

7. ∠5 & ∠4 are supplementary

7. c.

38. b. Transitive Property of

Congruence

39. a. Substitution Property of

Equality

40. d. Symmetric Property of

Congruence

41. c. Reflexive Property of

Equality

42. Proof reasons should be:

Statements Reasons

1. line m || line k 1. b.

Geometry: Parallel Lines ~18~ NJCTL.org

2. ∠3 ≅ ∠7 2. d.

3. ∠7 ≅ ∠5 3. a.

4. ∠3 ≅ ∠5 4. c.

43. Proof reasons should be:

Statements Reasons

1. line m || line k 1. c.

2. ∠1 ≅ ∠5 2. b.

3. m∠1 = m∠5 3. d.

4. ∠5 & ∠8 are supplementary

4. e.

5. m∠5 + m∠8 = 180

5. a.

6. m∠1 + m∠8 = 180

6. f.

7. ∠1 & ∠8 are supplementary

7. a.

44. 54°

45. 126°

46. 126°

47. 54°

48. 54°

49. 138°

50. 42°

51. 42°

52. 138°

53. x= 144°

54. x= 64° and y= 49/4

55. x=6; z=2

56. x=24, y=11; z=22/5

57. x=33; y=2

58. 44°

59. 107°

60. 29°

61. 29°

62. 136°

63. Segments 𝐴𝐷̅̅ ̅̅ and 𝐵𝐶 ̅̅ ̅̅ ̅are

parallel

64. Segments 𝑂𝑃̅̅ ̅̅ and 𝑅𝑆 ̅̅ ̅̅ are

parallel

65. None of these

66. x=9 and y=8 and z=7

67. x=8 and y=7

68. 62°

69. 118°

70. 118°

71. 62°

72. 62°

73. 144°

74. 36°

75. 36°

76. 144°

77. x=55°

78. x=86° and y=7

79. x=9; y=6; z=7

80. x=15; y=10; z=8

81. x=25; y=3

82. 41°

83. 106°

84. 33°

85. 33°

86. 129°

87. cannot be determined

88. Segments 𝑁𝐾̅̅̅̅̅ and 𝑀𝐿̅̅ ̅̅ are

parallel

89. Segments 𝑄𝑃̅̅ ̅̅ and 𝑇𝑆 ̅̅ ̅̅ are

parallel

90. x=6; y=12; z=7

91. x=18; y=7

92. See student work

93. made same side interior the

same

94. See student work

95. Made angles congruent that

should be supplementary.

96. see student work

97. see student work

Geometry: Parallel Lines ~19~ NJCTL.org

98. Part A: c & Part B: d

99. Part A: b & Part B: b

REVIEW 1. c 2. b 3. c 4. a 5. b 6. c 7. a 8. c 9. d 10. c 11. d

EXTENDED CONSTRUCTED RESPONSE 1.

Statements Reasons

∠1 ≅ ∠3 c. Given

𝑀𝑁̅̅ ̅̅ ̅ || 𝑃𝑄̅̅ ̅̅ c. Given

∠1 ≅ ∠2 b. Alternate Interior Angles Theorem

∠2≅∠3 a. Transitive Property of congruence

Statements Reasons

1. n || p, k || m 1. e

2. ∠2≅∠12 2. f

3. ∠12≅∠14 3. c

4. ∠2≅∠14 4. a

5. m∠2+m∠14 5. d

6. ∠13 & ∠14 are supplementary

6. g

7. m∠13 = m∠14 = 180° 7. b

8. m∠13 + m∠2 = 180° 8. h

9. ∠2 & ∠13 are supplementary

9. b

3. See student work