Angles and Perpendicular Lines

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 any any place other than the endpoints

82. Homework

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