1. Lesson 3.1 Lines and Angles
You will learn to …
* identify relationships between lines
* identify angles formed by transversals

2. If two lines are coplanar and do not intersect, then they are _______________
parallel lines

3. Parallel Lines B
AB || CD A D C r p
r || p

4. If two lines are NONcoplanar and do not intersect, then they are ____________
skew lines

5. If two lines intersect to form one right angle, then they are ___________________.
perpendicular lines
AB | CD
r | p

6. 1. t and u neither 2. t and s parallel 3. r and u perpendicular
Determine whether the lines are parallel, perpendicular, or neither. s
1. t and u t
neither
2. t and s
parallel r
3. r and u
perpendicular u

7. 4. t and u skew Determine whether the lines are intersecting or skew.

8. Postulate 13 Parallel Postulate Given a line and a point not on the line, then there is _________________ through the point parallel to the line.
exactly one line

9. Postulate 14 Perpendicular Postulate Given one line and one point, there is ________________ through the point perpendicular to the line.
exactly one line

10. a line that intersects two or more coplanar lines at different points
transversal –
a line that intersects two or more coplanar lines at different points no
transversal

11. b a c Identify the transversal: Line a Line b Line c Lines a and b
E) All 3 lines c

12. Special angles are formed when a transversal intersects 2 lines.
corresponding angles
Special angles are formed when a transversal intersects 2 lines. 1 1 2 2 4 4 3 3 5 5 6 6 7 7 8 8

13. alternate interior angles1 2 3 4 5 6 4 3 5 6 7 8

14. alternate exterior angles2 7 2 1 1 8 4 3 5 6 7 8

15. consecutive interior angles1 2 3 4 5 6 4 3 5 6 7 8
same-side interior angles

16. same-side exterior angles2 2 1 1 3 4 5 6 7 7 8 8

17. Describe the relationship between the given angles.1 2
5. 1 and 2 4 3
6. 3 and 4 5 6
7. 5 and 6
corresponding
alternate exterior
consecutive interior

18. Lesson 3.2 Proof & Perpendicular Lines
You will learn to …
* write different types of proofs
* prove results about perpendicular lines

19. 2. Which angles are adjacent? 1. Which angles are congruent?
1 and 4
2 and 4
1 and 3
2 and 3
1  2
3  4 2 1 3 4 2 1 3 4
3. How would the diagram change if adjacent angles were congruent?

20. Theorem 3.1 Perpendicular Lines Theorem If 2 lines intersect to form a linear pair of congruent angles, then the lines are _______________.
perpendicular

21. Perpendicular Lines Theorem
4. Is RP  ST ? How do you know? R
Yes, RP  ST. 1 2 S P T
m1 = m2 and
m1 + m2 = 180
m1 = 90 
m2 = 90 
Perpendicular Lines Theorem

22. Draw two adjacent acute
angles such that their uncommon sides are perpendicular.
What do you know about the two acute angles?

23. Theorem 3.2 Adjacent Complements Theorem If 2 sides of two adjacent acute angles are perpendicular, then the angles are ________________.
complementary

24. 5. AC  BD Find x. B
(6x + 4) + 20 = 90
x = 11 C A
20 P E
(6x + 4) D

25. Theorem Right Angles Theorem If 2 intersecting lines are _____________, then they form 4 right angles.
perpendicular

26. Determine whether enough information is given to conclude that the statement is true.
6. 1  2
7. 2  3
8. 3  4 1 c a 2 d
yes 4 3
yes b no

27. Look at your Ch 2 Celebration.
Paragraph Proofs
9. Given: AB = BC
Prove: ½ AC = BC A B C
AC = AB + BC by the Segment Addition Postulate. Since AB=BC, AC = BC + BC by substitution. By the Distributive Prop, AC = 2BC. ½ AC = BC by the Division Prop.
Look at your Ch 2 Celebration.

28. Look at your Ch 2 Celebration.
Paragraph Proofs
10. Given: 1 and 3 are a linear pair
2 and 3 are a
linear pair
Prove: m1 = m2 1 2 3
Since 1 & 3 and 2 & 3 are linear pairs, 1 & 3 are supplementary and 2 & 3 are supplementary by the Linear Pair Postulate. So, 1  2 by the Congruent Supplements Theorem. By definition of , m 1 = m 2
Look at your Ch 2 Celebration.

29. Flow Proofs 11. Given: 1 and 2 are a linear pair
Prove: m1 = m3 1 3 2

30. Flow Proofs 12. Given: 5  6 5 and 6 are a linear pair
Prove: j  k j 5 6 k

31. The best way for you to get better at writing proofs is to practice.
Don’t give up!

32. 1. Write a two-column proof of Theorem 3.1 Perpendicular Lines Theorem
Given:  1   2,  1 and  2 are a linear pair
Prove: g  h g h 1 2

33. 2. Write a two-column proof of Theorem 3
2. Write a two-column proof of Theorem 3.2 Adjacent Complements Theorem
Given: BA  BC
Prove: 1 and 2 are complementary A B 1 2 C

34. 3. Write a two-column proof of Theorem 3.3 4 Right Angles Theorem
Given: j  k ,
Prove: 2 is a right angle j 1 2 k

35. j 4 5 3 6 k 4. Given: j  k , 3 and 4 are complementary
Prove:  5   6 j 3 6 k 5 4

37. Lesson 3.3 Parallel Lines and Transversals
Students need a protractor and straight edge
You will learn to …
* prove and use results about parallel
lines and transversals
* use properties of parallel lines

38. If 2 parallel lines are cut by a transversal, then…
Postulate 15 & Theorems 3.4 – 3.6
If 2 parallel lines are cut by a transversal, then…
Use a straight edge to create 2 parallel lines cut by a transversal.

39. …corresponding angles are ______________.
congruent
Corresponding Angles Postulate

40. corresponding angles 1 2 3 4 5 6 7 8 1 5 2 6 4 8 3 7 1  5 3  7
2  6
4  8

41. …alternate interior angles are ______________.
congruent
Alternate Interior Angles Theorem

42. alternate interior angles3 4 5 6 4 5 3 6
3  6
4  5

43. … alternate exterior angles are _______________.
congruent
Alternate Exterior Angles Theorem

44. alternate exterior angles2 7 2 7 1 8 1 8
1  8
2  7

45. …consecutive interior angles are _______________.
supplementary
Consecutive Interior Angles Theorem

46. consecutive interior angles3 4 5 6
m3 + m5 = 180
m4 + m6 = 180

47. …same-side exterior angles are _______________.
supplementary
Same-Side Exterior Angles Theorem

48. same-side exterior angles1 2 7 8
m1 + m7 = 180
m2 + m8 = 180

49. Find the measure of the numbered angle.
110
1. m1 =
110 1
Corresponding s Postulate

50. Find the measure of the numbered angle.
100
2. m2 =
100 2
Alt. Ext. s Theorem

51. Find the measure of the numbered angle.
112
3. m3 =
112 3
Alt. Int. s Theorem

52. Find the measure of the numbered angle.
60
4. m4 =
120 4
Cons. Int. s Theorem

53. Find the measure of the numbered angle.
70
5. m5 =
110 5
Same-side Ext. s Theorem

54. 6. Find x.
125
(12x – 5)
12x – = 180
12x = 180
12x = 60
x = 5

55. 7. Find x.
100
(5x + 40)
5x + 40 = 100
5x = 60
x = 12

56. it is perpendicular to the other
Theorem 3.7  Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then … ________________________.
it is perpendicular to the other

57. m t n
If t  n
and n || m,
then t  m

58. A# 3.3
#28
Statements
1) j || k
2)  1   3
3)  2   3
4)  1   2
#29
Statements
1) p  q ; q || r
2)  1 is a right angle
3)  1   2
4)  2 is a right angle
5) p  r

60. Lesson 3.4 & 3.5 Parallel Lines
You will learn to …
* prove that two lines are parallel
* use properties of parallel lines

61. What is the converse of a conditional if-then statement?

62. Write the converse of the statement.
1. If two parallel lines are cut by a transversal, then corresponding angles are congruent.
If corresponding angles are congruent, then the two lines cut by the transversal are parallel.

63. Postulate 16 Corresponding Angles Converse
If corresponding angles are___, then… 1 5 2 6 3 7 4 8

64. Theorem 3.8 – Alternate Interior s Converse
If alternate interior angles are___, then…  5 6 3 4

65. Theorem 3.9 Consecutive Interior s Converse
If same-side interior angles are
supplementary
, then… 5 6 3 4

66. Theorem 3.10 – Alternate Exterior Angles Converse
If alternate exterior angles are___, then…  1 2 7 8

67. Same-side Exterior s Converse
If same-side exterior angles are
supplementary
, then… 1 2 7 8

68. …the 2 lines cut by the transversal are parallel

69. || lines || lines IF THEN angles IF THEN angles Postulate and Theorems
Converse
|| lines
IF THEN
angles

70. 2. Can you prove that n || m ? Explain.
112 n
112 m
Yes, Corresponding s Converse

71. 3. Can you prove that n || m ? Explain.
78 n
78 m
Yes, Alternate Ext. s Converse

72. 4. Can you prove that n || m ? Explain.
72 n
108 m
Yes, Consecutive Interior s Converse

73. 5. Can you prove that n || m ? Explain.
102
102
Yes, Alternate Interior s Converse

74. 6. Can you prove that n || m ? Explain.
123 n
47 m NO
 180

75. 7. Can you prove that n || m ? Explain.
100
100 n m NO

76. If p || k and n || k, then…?
p || n p k n

77. If r || s and s || t, then ____
3 Parallel Lines Theorem
Theorem If 2 lines are parallel to the same line, then they are ____________ to each other.
parallel
r || t.
If r || s and s || t, then ____

78. If p  k and n  k, then…?
p || n p n k

79.   || Theorem
Theorem If 2 lines are perpendicular to the same line, then they are _________ to each other.
parallel
r || t
If r  s and t  s, then ____

80. A# 3.4
#30
Statements
1)  4   5
2)  4 &  6 are vertical angles
3)  4   6
4)  5   6
5) g || h

81. A# 3.4
#32
Statements
1)  B   BEA
2)  BEA   CED
3)  CED   C
4)  B   C
5) AB || CD

82. A# 3.4
#34
Statements
1) m  7 = 125°; m  8 = 55°
2) m  7 + m  8 = 125° + m  8
3) m  7 + m  8 = 125° + 55°
4) m  7 + m  8 = 180°
5)  7 &  8 are supplementary
6) j || k

83. Students need scissors, glue, and 2 sheets of paper.
Do Practice Proofs…
Students need scissors, glue, and 2 sheets of paper.

85. Lesson 3.6 & 3.7 Parallel and Perpendicular Lines
You will learn to …
* find slopes of lines and use slope to
identify parallel lines and
perpendicular lines
* write equation of parallel lines
* write equations of perpendicular lines

86. y2 – y1 x2 – x1 Slope = rise run = The slope of a line is the
ratio of the
vertical change (rise) to the horizontal change (run).
y2 – y1
x2 – x1
Slope =
rise
run =

87. m = -1 m = 0 m = undefined 1. (-5,7) (-2,4) 2. (3, -2) (-5, -2)
Find the slope of the line that passes through the given points.
m = -1
1. (-5,7) (-2,4)
m = 0
2. (3, -2) (-5, -2)
3. (-6, 2) (-6, -2)
m = undefined

88. All vertical lines are parallel. All horizontal lines are parallel.
Postulate 17 Slopes of Parallel Lines Lines are parallel if and only if they have the same slope.
All vertical lines are parallel. All horizontal lines are parallel.

89. vertical lines v
horizontal lines h v
x = #
y = #
Slope is 0
Slope is undefined

90. slope-intercept form
y = mx + b
y–intercept
(0,b)
slope

91. m = - 5 m = ½ 4. y = -5x + 14 5. 2x – 4y = -3 – 4y = – 2x – 3
Identify the slope of the line.
m = - 5
4. y = -5x + 14
5. 2x – 4y = -3
– 4y = – 2x – 3 -4
m = ½
y = ½ x + ¾

92. y = -2x + 3 y = - 5 6. slope = - 2 y-int = 3 7. slope = 0 y-int = -5
Write the equation of the line with the given slope and y-intercept.
6. slope = - 2
y-int = 3
y = -2x + 3
7. slope = 0
y-int = -5
y = - 5

93. Write the equation of the line that has a y-intercept of -7 and is parallel to the given line.
8. y = - ½ x + 10
- ½
m =
y = - ½ x – 7

94. point-slope form
y – y1 = m (x – x1)

95. y – y1 = m (x – x1) y - 3 = 5 (x - 2) y - 3 = 5x - 10 y = 5x - 7
9. Use the point-slope form to write the equation of the line through the point (2, 3) that has a slope of 5.
y – y1 = m (x – x1)
y - 3 = 5 (x - 2)
y - 3 = 5x - 10
y = 5x - 7

96. y – y1 = m (x – x1) y - -4 = - ½ (x - -2) y + 4 = - ½ (x + 2)
10. Write the equation of the line through the point (-2, -4) that is parallel to y = - ½ x + 5.
y – y1 = m (x – x1)
y - -4 = - ½ (x - -2)
y + 4 = - ½ (x + 2)
y + 4 = - ½ x - 1
y = - ½ x - 5

97. Opposite & Reciprocals
Postulate 18 Slopes of Perpendicular Lines Lines are perpendicular if and only if the product of their slopes is -1.
Opposite & Reciprocals

98. 1 m = 5 m = - 2 m = undefined 11. y = -5x + 14 12. y = ½ x - 7
Identify the slope of the line that is perpendicular to the given line.
m = 1 5
11. y = -5x + 14
12. y = ½ x - 7
m = - 2
13. y = - 8
m = undefined

99. y = - 1/3 x + 7 14. y = 3x – 2 (9,4) m = - 1/3 ? y – 4 = - 1/3 (x – 9)
Find the equation of the line that is perpendicular to the given line and passes through the given point.
14. y = 3x – 2 (9,4)
m =
- 1/3 ?
y – 4 = - 1/3 (x – 9)
y – 4 = - 1/3 x + 3
y = - 1/3 x + 7

100. y = - 7x + 25 15. y = 1/7x – 11 (5, -10) m = -7 ? y + 10 = - 7 (x – 5)
Find the equation of the line that is perpendicular to the given line and passes through the given point.
15. y = 1/7x – 11 (5, -10)
m = -7 ?
y + 10 = - 7 (x – 5)
y + 10 = - 7x + 35
y = - 7x + 25