Let x = the measure of the smaller angle Then the measure of the larger angle is x + 40 We know: x + (x + 40) = 180 2x + 40 = 180 2x = 140 x = 70 Therefore, the measures of the angles are: Smaller angle: 70° Larger angle: 70° + 40° = 110°

1. SECTION 5-2
Angles and Perpendicular Lines

2. Essential Questions
How do you identify and use perpendicular lines?
How do you identify and use angle relationships?
Where you’ll see this:
Health, physics, paper folding, navigation

3. Vocabulary
1. Ray:
2. Opposite Rays:
3. Angle:
4. Vertex:
5. Degrees:
6. Complementary Angles:
7. Supplementary Angles:

4. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays:
3. Angle:
4. Vertex:
5. Degrees:
6. Complementary Angles:
7. Supplementary Angles:

5. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle:
4. Vertex:
5. Degrees:
6. Complementary Angles:
7. Supplementary Angles:

6. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle: A figure made up of two rays that share an endpoint
4. Vertex:
5. Degrees:
6. Complementary Angles:
7. Supplementary Angles:

7. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle: A figure made up of two rays that share an endpoint
4. Vertex: The point that is shared by the rays of an angle
5. Degrees:
6. Complementary Angles:
7. Supplementary Angles:

8. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle: A figure made up of two rays that share an endpoint
4. Vertex: The point that is shared by the rays of an angle
5. Degrees: The size measurement of an angle
6. Complementary Angles:
7. Supplementary Angles:

9. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle: A figure made up of two rays that share an endpoint
4. Vertex: The point that is shared by the rays of an angle
5. Degrees: The size measurement of an angle
6. Complementary Angles: Two angles that add up to 90 degrees
7. Supplementary Angles:

10. Vocabulary
1. Ray: Part of a line that begins at an endpoint and goes on forever in
one direction
2. Opposite Rays: Two rays that together form a straight line
3. Angle: A figure made up of two rays that share an endpoint
4. Vertex: The point that is shared by the rays of an angle
5. Degrees: The size measurement of an angle
6. Complementary Angles: Two angles that add up to 90 degrees
7. Supplementary Angles: Two angles that add up to 180 degrees

11. Vocabulary
8. Adjacent Angles:
9. Congruent Angles:
10. Perpendicular Lines:
11. Vertical Angles:
12. Bisector of an Angle:

12. Vocabulary
8. Adjacent Angles: Two angles that share a side and vertex but no other
points
9. Congruent Angles:
10. Perpendicular Lines:
11. Vertical Angles:
12. Bisector of an Angle:

13. Vocabulary
8. Adjacent Angles: Two angles that share a side and vertex but no other
points
9. Congruent Angles: Two or more angles with the same measure
10. Perpendicular Lines:
11. Vertical Angles:
12. Bisector of an Angle:

14. Vocabulary
8. Adjacent Angles: Two angles that share a side and vertex but no other
points
9. Congruent Angles: Two or more angles with the same measure
10. Perpendicular Lines: Lines that intersect at a 90 degree angle
11. Vertical Angles:
12. Bisector of an Angle:

15. Vocabulary
8. Adjacent Angles: Two angles that share a side and vertex but no other
points
9. Congruent Angles: Two or more angles with the same measure
10. Perpendicular Lines: Lines that intersect at a 90 degree angle
11. Vertical Angles: When two lines/segments intersect, the two angles that
are not adjacent to each other; they are congruent
12. Bisector of an Angle:

16. Vocabulary
8. Adjacent Angles: Two angles that share a side and vertex but no other
points
9. Congruent Angles: Two or more angles with the same measure
10. Perpendicular Lines: Lines that intersect at a 90 degree angle
11. Vertical Angles: When two lines/segments intersect, the two angles that
are not adjacent to each other; they are congruent
12. Bisector of an Angle: A ray that divides an angle into two equal angles

17. Ray
Part of a line that begins at an endpoint and goes on forever in one
direction

18. Ray
Part of a line that begins at an endpoint and goes on forever in one
direction
A B

19. Opposite Rays
Two rays that together form a straight line

20. Opposite Rays
Two rays that together form a straight line
A B

21. Opposite Rays
Two rays that together form a straight line
C A B

22. Angle
A figure made up of two rays that share an endpoint

23. Angle
A figure made up of two rays that share an endpoint
C
A
B

24. Vertex
The point that is shared by the rays of an angle
C
A
B

25. Vertex
The point that is shared by the rays of an angle
C
A
B

26. Degrees
The size measurement of an angle
C
A
B

27. Degrees
The size measurement of an angle
C
A
B
http://www.chutedesign.co.uk/design/protractor/protractor.gif

28. Complementary Angles
Two angles that add up to 90 degrees

29. Complementary Angles
Two angles that add up to 90 degrees
D
A
C
B

30. Supplementary Angles
Two angles that add up to 180 degrees

31. Supplementary Angles
Two angles that add up to 180 degrees
C
D
B
A

32. Adjacent Angles
Two angles that share a side and vertex but no other points
C
D
B
A

33. Congruent Angles
Two or more angles with the same measure

34. Congruent Angles
Two or more angles with the same measure
A
M N

35. Perpendicular Lines
Lines that intersect at a 90 degree angle

36. Perpendicular Lines
Lines that intersect at a 90 degree angle
g
h

37. Perpendicular Lines
Lines that intersect at a 90 degree angle
g
r
h s

38. Perpendicular Lines
Lines that intersect at a 90 degree angle
g
r
h s
perpendicular

39. Perpendicular Lines
Lines that intersect at a 90 degree angle
g
r
h s
perpendicular
NOT perpendicular

40. Vertical Angles
When two lines/segments intersect, the two angles that are not
adjacent to each other; they are congruent

41. Vertical Angles
When two lines/segments intersect, the two angles that are not
adjacent to each other; they are congruent
B
A
E
C
D

42. Vertical Angles
When two lines/segments intersect, the two angles that are not
adjacent to each other; they are congruent
B
A
E
C
D

43. Vertical Angles
When two lines/segments intersect, the two angles that are not
adjacent to each other; they are congruent
B
A
E
C
D

44. Bisector of an Angle
A ray that divides an angle into two equal angles

45. Bisector of an Angle
A ray that divides an angle into two equal angles
N
A
D

46. Bisector of an Angle
A ray that divides an angle into two equal angles
N
E
A
D

47. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.

48. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle:

49. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: Larger angle:

50. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle:

51. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40

52. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180

53. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180
2x + 40 =180

54. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180
2x + 40 =180
−40 −40

55. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180
2x + 40 =180
−40 −40
2x =140

56. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180
2x + 40 =180
−40 −40
2x =140
2 2

57. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180
2x + 40 =180
−40 −40
2x =140
2 2
x = 70

58. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180 70 + 40
2x + 40 =180
−40 −40
2x =140
2 2
x = 70

59. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180 70 + 40 =110
2x + 40 =180
−40 −40
2x =140
2 2
x = 70

60. Example 1
The larger of the two supplementary angles measures 40° more than
the smaller. Find the measure of each angle.
Smaller angle: x Larger angle: x + 40
x + x + 40 =180 70 + 40 =110
2x + 40 =180
−40 −40
2x =140 The angles are 70° and 110°
2 2
x = 70

61. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.

62. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B
m∠AQE = 4 0°
E
F
Q
C
D

63. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40°
m∠AQE = 4 0°
E
F
Q
C
D

64. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0°
E
F
Q
C
D

65. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0°
E
F
Q
C
D

66. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40°
E
F
Q
C
D

67. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
F
Q
C
D

68. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
F
Q
C
D

69. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40°
F
Q
C
D

70. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40° (vertical angles)
F
Q
C
D

71. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40° (vertical angles)
F
Q
C
D

72. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40° (vertical angles)
F
Q
m∠BQD = m∠BQF + m∠FQD
C
D

73. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40° (vertical angles)
F
Q
m∠BQD = m∠BQF + m∠FQD
C
= 40° + 40° = 80°
D

74. Example 2
Draw the figure where AD and CB intersect at Q. EF bisects ∠AQC.
m∠AQE = 40°. Find m∠BQD.
A
B m∠FQD = 40° (vertical angles)
m∠AQE = 4 0° m∠EQC = 40° (angle bisector)
E
m∠BQF = 40° (vertical angles)
F
Q
m∠BQD = m∠BQF + m∠FQD
C
= 40° + 40° = 80°
D
m∠BQD = 80°

75. Example 3
If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C
supplementary angles? Explain.

76. Example 3
If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C
supplementary angles? Explain.
20° + 60° +100° =180°?

77. Example 3
If m∠A = 20°, m∠B = 60°, and m∠C = 100°, are angles A, B, and C
supplementary angles? Explain.
20° + 60° +100° =180°?
Yes. Since the sum of the three angles is 180°, they are supplementary.

78. Example 4
Are angles 1 and 2 adjacent? Explain.
1 2

79. Example 4
Are angles 1 and 2 adjacent? Explain.
1 2
These are not adjacent. In order to be adjacent, the two angles must
share both the vertex and a side.

80. Example 5
Are ∠AEB and∠CED vertical angles? Explain.
B
A
E C
D

81. Example 5
Are ∠AEB and∠CED vertical angles? Explain.
B
A
E C
D
There are no vertical angles here, as none of the angles are formed by
segments intersecting at any place other than the endpoints

82. Problem Set

83. Problem Set
p. 198 #1-24
“No problem is too small or too trivial if we can really do something
about it.” - Richard Feynman