parallel lines chapter problems - content.sandbox...
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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