Download Parallel and Perpendicular Lines

Parallel Lines Chapter Problems
Lines: Intersecting, parallel & skew
Class Work – Use image 1
̅̅̅̅:
1. Name all segments parallel to 𝐺𝐻
̅̅̅̅
2. Name all segments skew to 𝐺𝐻 :
3. Name all segments intersecting with ̅̅̅̅
𝐺𝐻 :
̅̅̅̅ and 𝐵𝐴
̅̅̅̅ coplanar? Explain your answer.
4. Are segments 𝐺𝐻
5. Are segments ̅̅̅̅
𝐺𝐻 and ̅̅̅̅
𝐵𝐹 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 ̅̅̅̅
𝐹𝐸 :
̅̅̅̅ :
12. Name all segments skew to 𝐹𝐸
13. Name all segments intersecting with ̅̅̅̅
𝐹𝐸 :
̅̅̅̅
̅̅̅̅
14. Are segments 𝐹𝐸 and 𝐶𝐷coplanar? Explain your answer.
̅̅̅̅ and 𝐻𝐷
̅̅̅̅ coplanar? Explain your answer.
15. Are segments 𝐹𝐸
Image 1
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, sameside 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
Geometry – Parallel Lines
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Classify each pair of angles as alternate interior, alternate exterior, same-side interior, sameside 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
36. If DE = FG, then FG = DE.
a) Substitution Property of Equality
34. If m∠A = m∠B and m∠B = m∠C, then
b) Transitive Property of Equality
m∠A = m∠C.
c) Reflexive Property of Equality
35. If x + y = 9 and y = 5, then x + 5 = 9.
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
1. line m || line k
2. ∠2 ≅ ∠6
Reasons
1.
2.
3. ∠6 ≅ ∠8
3.
4. ∠2 ≅ ∠8
4.
nce
nding
Geometry – Parallel Lines
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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
Statements
1. line m || line k
2. ∠1 ≅ ∠5
3. m∠1 = m∠5
4. ∠1 & ∠4 are supplementary
5. m∠1 + m∠4 = 180
6. m∠5 + m∠4 = 180
7. ∠5 & ∠4 are supplementary
Reasons
1.
2.
3.
4.
5.
6.
7.
a)
b)
c)
d)
e)
f)
Parallel Lines & Proofs
Homework
For #39-42 match the description on the left to
39. ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
40. If bc = 77 and b = 11, then 11c = 77.
41. If ∠P ≅ ∠M, then ∠M ≅ ∠P.
42. QR = QR
Reasons Bank
Angles that form a linear pair are
supplementary.
Substitution Property of Equality
Definition of supplementary angles
If 2 parallel lines are cut by a
transversal, then the corresponding
angles are congruent.
Definition of congruent angles
Given
the name of the property on the right.
a) Substitution Property of Equality
b) Transitive Property of Congruence
c) Reflexive Property of Equality
d) Symmetric Property of Congruence
PARCC type question:
Geometry – Parallel Lines
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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
1. line m || line k
2. ∠3 ≅ ∠7
3. ∠7 ≅ ∠5
4. ∠3 ≅ ∠5
Reasons
1.
2.
3.
4.
a)
b)
c)
d)
Geometry – Parallel Lines
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Reasons Bank
Vertical Angles are congruent.
Given
Transitive Property of Congruence
If 2 parallel lines are cut by a transversal,
then the corresponding angles are
congruent.
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
Statements
1. line m || line k
2. ∠1 ≅ ∠5
3. m∠1 = m∠5
4. ∠5 & ∠8 are
supplementary
5. m∠5 + m∠8 = 180
6. m∠1 + m∠8 = 180
7. ∠1 & ∠8 are
supplementary
Reasons
1.
2.
3.
4.
5.
6.
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
Properties of Parallel Lines
Classwork
Use the given diagram to answer problems #45-53.
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=
Geometry – Parallel Lines
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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.
Geometry – Parallel Lines
58.
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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
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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.
81.
Geometry – Parallel Lines
79.
80.
82.
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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.
D
C
124°
124°
A
B
90.
89.
91.
Geometry – Parallel Lines
92.
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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
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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
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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.
A
j
B
C
Part A:
A
j
C
F
G B
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.
Geometry – Parallel Lines
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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.
J
p
H
K
M
Part A:
N
p
H
J
K
M
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.
Geometry – Parallel Lines
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Parallel Lines Review
Multiple Choice
1. Name the segment parallel to ̅̅̅̅
𝐺𝐻 and skew to ̅̅̅̅
𝐸𝐴.
̅̅̅̅
a. 𝐹𝐵
̅̅̅̅
b. 𝐷𝐴
̅
c. 𝐽𝐼
̅̅̅̅
d. 𝐻𝐷
̅̅̅̅ and skew to 𝐸𝐼.
̅̅̅̅
2. Name the segment parallel to 𝐵𝐶
̅̅̅̅
a. 𝐹𝐵
b. ̅̅̅̅
𝐷𝐴
̅
c. 𝐽𝐼
̅̅̅̅
d. 𝐻𝐷
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
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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; ̅̅̅̅̅
𝑀𝑁 || ̅̅̅̅
𝑃𝑄
Prove: ∠2≅∠3
M
1
3
P
Statements
1. ∠1 ≅ ∠3
̅̅̅̅
̅̅̅̅̅ || 𝑃𝑄
2. 𝑀𝑁
3. ∠1 ≅ ∠2
4. ∠2≅∠3
Geometry – Parallel Lines
Reasons
1.
2.
3.
4.
N
2
Q
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
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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
1. n || p, k || m
2. ∠2 ≅ ∠12
3. ∠12 ≅ ∠14
4. ∠2 ≅ ∠14
5. m∠2 = m∠14
6. m∠13 & m∠14 are
supplementary
7. m∠13 + m∠14 = 180°
8. m∠13 + m∠2 = 180°
9. ∠2 &∠13 are supplementary
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
9.
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
3. Using a compass and straightedge, construct parallel lines. You can use any method of
your choice.
Geometry – Parallel Lines
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Answers
̅ ,𝐹𝐸
̅̅̅̅ , ̅̅̅̅
1. Segments 𝐽𝐼
𝐵𝐴, ̅̅̅̅
𝐶𝐷
̅̅̅̅
̅̅̅ ,𝐼𝐸
̅̅̅ , 𝐹𝐵
̅̅̅̅, 𝐸𝐴
Segments 𝐽𝐹
̅̅̅, 𝐺𝐹
̅̅̅̅ , 𝐺𝐶
̅̅̅̅ ,
2. Segments 𝐺𝐽
̅̅̅̅ , 𝐻𝐸
̅̅̅̅ , 𝐻𝐷
̅̅̅̅
𝐻𝐼
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
̅̅̅̅ , 𝐵𝐴
̅̅̅̅, 𝐶𝐷
̅̅̅̅
̅ , 𝐺𝐻
10. Segments 𝐽𝐼
̅̅̅, 𝐶𝐺
̅̅̅̅ , 𝐷𝐻
̅̅̅̅ , 𝐺𝐽
̅̅̅̅
11. Segments 𝐻𝐼
̅̅̅̅ , 𝐹𝐵
̅̅̅̅, 𝐹𝐽
̅̅̅, ̅̅̅̅
̅̅̅̅,
12. Segments 𝐹𝐺
𝐸𝐴, 𝐹𝐵
̅̅̅̅
𝐸𝐻 , ̅̅̅
𝐸𝐼
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
Geometry – Parallel Lines
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. b.
2. ∠2 ≅ ∠6
3. c.
3. ∠6 ≅ ∠8
4. a.
4. ∠2 ≅ ∠8
37. Proof reasons should be:
Statements
Reasons
1. line m || line k
1. f.
2. d.
2. ∠1 ≅ ∠5
3. e.
3. m∠1 = m∠5
4. a.
4. ∠1 & ∠4 are
supplementary
5. c.
5. m∠1 + m∠4 =
180°
6. b.
6. m∠5 + m∠4 =
180°
7. c.
7. ∠5 & ∠4 are
supplementary
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.
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NJCTL.org
2. ∠3 ≅ ∠7
3. ∠7 ≅ ∠5
4. ∠3 ≅ ∠5
2. d.
3. a.
4. c.
43. Proof reasons should be:
Statements
Reasons
1. line m || line k
1. c.
2. b.
2. ∠1 ≅ ∠5
3. d.
3. m∠1 = m∠5
4. e.
4. ∠5 & ∠8 are
supplementary
5. a.
5. m∠5 + m∠8 =
180
6. f.
6. m∠1 + m∠8 =
180
7. a.
7. ∠1 & ∠8 are
supplementary
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 ̅̅̅̅
𝐴𝐷 and ̅̅̅̅̅
𝐵𝐶 are
parallel
Geometry – Parallel Lines
NJCTL.org
̅̅̅̅and 𝑅𝑆
̅̅̅̅are
64. Segments 𝑂𝑃
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 ̅̅̅̅̅
𝑁𝐾 and ̅̅̅̅
𝑀𝐿are
parallel
89. Segments ̅̅̅̅
𝑄𝑃 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
~18~
98. Part A: c & Part B: d
99. Part A: b & Part B: b
3. See student work
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
∠1 ≅ ∠3
̅̅̅̅̅
𝑀𝑁 || ̅̅̅̅
𝑃𝑄
∠1 ≅ ∠2
∠2≅∠3
Reasons
c. Given
c. Given
b. Alternate
Interior
Angles
Theorem
a. Transitive
Property of
congruence
Statements
1. n || p, k || m
2. ∠2≅∠12
3. ∠12≅∠14
4. ∠2≅∠14
5. m∠2+m∠14
6. ∠13 & ∠14 are
supplementary
7. m∠13 = m∠14 = 180°
8. m∠13 + m∠2 = 180°
9. ∠2 & ∠13 are
supplementary
Geometry – Parallel Lines
NJCTL.org
Reasons
1. e
2. f
3. c
4. a
5. d
6. g
7. b
8. h
9. b
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