1. All About Angles

2. Two rays that meet at a point and extend indefinitely form an
Two rays that meet at a point and extend indefinitely form an . The two rays are the of the angle. The point at which they meet is the of the angle.

3. Two rays that meet at a point and extend indefinitely form an angle
Two rays that meet at a point and extend indefinitely form an angle. The two rays are the sides of the angle. The point at which they meet is the vertex of the angle.

4. An angle separates the plane into the region, the region, and the .

5. An angle separates the plane into the interior region, the exterior region, and the angle itself.

6. Angles are measured in using a protractor.

7. Angles are measured in degrees using a protractor.
45 0

8. Angles with equal measures are called .

9. Angles with equal measures are called congruent angles.

10. Angles can be named by:
The vertex point if there are no other angles at that vertex that could be confused.
Three letters with the vertex as the center letter and the other letters representing points from each side.
A small number if one is given in the angle.

11. Angles can be named by: <R R
The vertex point if there are no other angles at that vertex that could be confused.
Three letters with the vertex as the center letter and the other letters representing points from each side.
A small number if one is given in the angle.
<R R

12. Angles can be named by: <R R
The vertex point if there are no other angles at that vertex that could be confused.
Three letters with the vertex as the center letter and the other letters representing points from each side.
A small number if one is given in the angle.
<R R

13. Angles can be named by: <KRB K R B
The vertex point if there are no other angles at that vertex that could be confused.
Three letters with the vertex as the center letter and the other letters representing points from each side.
A small number if one is given in the angle.
<KRB K R B

14. <1 Angles can be named by: 1
The vertex point if there are no other angles at that vertex that could be confused.
Three letters with the vertex as the center letter and the other letters representing points from each side.
A small number if one is given in the angle.
<1 1

15. Examples: Skip question 1. We will come back to it later.
#2. Why can this angle not be named <P?
#3. Name ALL the angles:
Skip question 4.

16. Examples:
#3. Name ALL the angles:
<MPN
<NPO
<MPO

17. Angle Classification
angles measure more than 0 degrees but less than 90 degrees

18. Angle Classification
Acute angles measure more than 0 degrees but less than 90 degrees

19. Angle Classification angles measure exactly 90 degrees.
lines form right angles.

20. Angle Classification Right angles measure exactly 90 degrees.
Perpendicular lines form right angles.

21. Angle Classification
angles measure greater than 90 degrees but less than 180 degrees.

22. Angle Classification
Obtuse angles measure greater than 90 degrees but less than 180 degrees.

23. Angle Classification
A angle (line) has a measure of 180 degrees.

24. Angle Classification
A straight angle (line) has a measure of 180 degrees.

25. When we want to say what the measure of an angle is, for example <ABC, we write m<ABC = 45 degrees

26. Example #5: Classify the Angles

27. Example #5: Classify the Angles
Right Angle

28. Example #5: Classify the Angles

29. Example #5: Classify the Angles
<ABC is an obtuse angle
<ABD is an acute angle

30. Protractor Postulate
Given with point O between A and B. consider ray OA and ray OB and any other rays that can be drawn with O as the endpoint on one side of line AB. These rays can be paired with the numbers from zero degrees to 180 degrees such that
Ray OA is paired with zero
Ray OB is paired with 180 degrees
If ray OC is paired with c degrees and ray OD is paired with d degrees, then the m<COD is the absolute value of the difference of c degrees and d degrees

31. Example #6: Find m<COD
Positions:
Ray OA at 0 degrees
Ray OB at 180 degrees
Ray OC at 75 degrees
Ray OD at 140 degrees

32. Example #6: Find m<COD
Positions:
Ray OA at 0 degrees
Ray OB at 180 degrees
Ray OC at 75 degrees
Ray OD at 140 degrees 75
140
180

33. Example #6: Find m<COD
Positions:
Ray OA at 0 degrees
Ray OB at 180 degrees
Ray OC at 75 degrees
Ray OD at 140 degrees
Big # - Small # = Total 75
140
180

34. Example #6: Find m<COD
Positions:
Ray OA at 0 degrees
Ray OB at 180 degrees
Ray OC at 75 degrees
Ray OD at 140 degrees
Big # - Small # = Total
140 – 75 = Total 75
140
180

35. Example #6: Find m<COD
Positions:
Ray OA at 0 degrees
Ray OB at 180 degrees
Ray OC at 75 degrees
Ray OD at 140 degrees
Big # - Small # = Total
140 – 75 = Total
65 = Total
m<COD = 65 degrees 75
140
180

36. Angle Addition Postulate
If D is in the interior of <ABC then
m<ABD + m<DBC = m<ABC

37. Example #7: Find m<ABC
m<ABD = 62 degrees
m<DBC = 31 degrees

38. Example #7: Find m<ABC
m<ABD = 62 degrees
m<DBC = 31 degrees
m<ABD + m<DBC = m<ABC
620
310

39. Example #7: Find m<ABC
m<ABD = 62 degrees
m<DBC = 31 degrees
m<ABD + m<DBC = m<ABC
= x
620
310

40. Example #7: Find m<ABC
m<ABD = 62 degrees
m<DBC = 31 degrees
m<ABD + m<DBC = m<ABC
= x
93 = x
620
310

41. Example #7: Find m<ABC
m<ABD = 62 degrees
m<DBC = 31 degrees
m<ABD + m<DBC = m<ABC
= x
93 = x
m<ABC = 93
620
310

42. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
560
900

43. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
m<ABD + m<DBC = m<ABC
560
900

44. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
m<ABD + m<DBC = m<ABC
56 + x = 90
560
900

45. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
m<ABD + m<DBC = m<ABC
56 + x = 90 x=
560
900

46. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
m<ABD + m<DBC = m<ABC
56 + x = 90 x=
X = 34
560
900

47. Example #8: Find m<DBC
m<ABC = 90 degrees
m<ABD = 56 degrees
m<ABD + m<DBC = m<ABC
56 + x = 90 x=
X = 34
m<DBC = 34
560
900

48. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
880

49. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
m<ABD + m<DBC = m<ABC x0 x0
880

50. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
m<ABD + m<DBC = m<ABC
x + x = 88 x0 x0
880

51. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
m<ABD + m<DBC = m<ABC
x + x = 88
2x= 88 x0 x0
880

52. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
m<ABD + m<DBC = m<ABC
x + x = 88
2x= 88
X = 44 x0 x0
880

53. Example #9: Find m<ABD & m<DBC
m<ABC = 88 degrees
m<ABD + m<DBC = m<ABC
x + x = 88
2x= 88
X = 44
m<DBC = 44
m<ABD = 44 x0 x0
880

54. Angle Relationships
- coplanar angles that have a common vertex and one common side, but NO common interior points

55. Angle Relationships
Adjacent Angles - coplanar angles that have a common vertex and one common side, but NO common interior points

56. Angle Relationships
- Two non adjacent angles formed by two intersection lines

57. Angle Relationships <1 & <2 <3 & <4
Vertical Angles - Two non adjacent angles formed by two intersection lines
<1 & <2
<3 & <4

58. Angle Relationships
- Two adjacent angles whose non common sides are two rays going in opposite directions

59. Angle Relationships <1 & <2
Linear Pair - Two adjacent angles whose non common sides are two rays going in opposite directions
<1 & <2

60. Angle Relationships
- Sum of the measures of the two angles is 90 degrees

61. Angle Relationships
Complementary Angles - Sum of the measures of the two angles is 90 degrees

62. Angle Relationships
Complementary Angles - Sum of the measures of the two angles is 90 degrees
600
300

63. Angle Relationships
- Sum of the measures of the two angles is 180 degrees.
If two angles form a linear pair, they are .

64. Angle Relationships
Supplementary Angles- Sum of the measures of the two angles is 180 degrees.
If two angles form a linear pair, they are supplementary.
1200
600

65. Angle Relationships
Supplementary Angles- Sum of the measures of the two angles is 180 degrees.
If two angles form a linear pair, they are supplementary.
1200
600

66. Angle Theorems Vertical angles are .
If two angles are supplementary to the same angle or to congruent angles, they are .
If two angles are complementary to the same angle or to congruent angles, they are .

67. Angle Theorems Vertical angles are congruent.
If two angles are supplementary to the same angle or to congruent angles, they are congruent.
If two angles are complementary to the same angle or to congruent angles, they are congruent.

68. Example #10: Find x
m<AOC = 16x – 20
m<BOD = 13x + 7

69. Example #10: Find x What type of angles are these? m<AOC = 16x – 20
m<BOD = 13x + 7
What type of angles are these?
16x - 20
13x + 7

70. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
16x - 20
13x + 7

71. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
m<AOC = m<BOD
16x - 20
13x + 7

72. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
m<AOC = m<BOD
16x – 20 = 13x +7
16x - 20
13x + 7

73. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
m<AOC = m<BOD
16x – 20 = 13x +7
3x – 20 = 7
16x - 20
13x + 7

74. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
m<AOC = m<BOD
16x – 20 = 13x +7
3x – 20 = 7
3x = 27
16x - 20
13x + 7

75. Vertical angles are congruent
Example #10: Find x
Vertical angles are congruent
m<AOC = 16x – 20
m<BOD = 13x + 7
m<AOC = m<BOD
16x – 20 = 13x +7
3x – 20 = 7
3x = 27
X = 9
16x - 20
13x + 7

76. Example #11: Find m<AOB & m<AOC
m<AOB = 4x + 15
m<AOC = 3x + 25

77. Example #11: Find m<AOB & m<AOC
m<AOB = 4x + 15
m<AOC = 3x + 25
What type of angles are these?
3x + 25
4x + 15

78. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
3x + 25
4x + 15

79. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
3x + 25
4x + 15

80. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
3x + 25
4x + 15

81. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
3x + 25
4x + 15

82. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
3x + 25
4x + 15

83. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
3x + 25
4x + 15

84. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
3x + 25
4x + 15

85. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
3x + 25
4x + 15

86. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
3x + 25
4x + 15

87. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
4x + 15

88. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
m<AOC = 3x +25
4x + 15

89. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
m<AOC = 3x +25
m<AOC = 3(20)+25
4x + 15

90. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
m<AOC = 3x +25
m<AOC = 3(20)+25
m<AOC =
4x + 15

91. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
m<AOC = 3x +25
m<AOC = 3(20)+25
m<AOC =
m<AOC=85 degrees
4x + 15

92. Example #11: Find m<AOB & m<AOC
Linear Pair
m<AOB = 4x + 15
m<AOC = 3x + 25
m<AOB + m<AOC = 180
4x x + 25 = 180
7x +40 = 180
7x = 140
X = 20
m<AOB = 4x +15
m<AOB = 4(20)+15
m<AOB = 80+15
m<AOB=95 degrees
3x + 25
m<AOC = 3x +25
m<AOC = 3(20)+25
m<AOC =
m<AOC=85 degrees
4x + 15

93. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20

94. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X + m<Y = 90

95. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X + m<Y = 90
3x x +20 = 90

96. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90

97. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63

98. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
X = 7

99. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7

100. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7

101. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7

102. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7

103. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7
m<Y = 6x +20

104. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7
m<Y = 6x +20
m<Y = 6(7) + 20

105. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7
m<Y = 6x +20
m<Y = 6(7) + 20
m<Y =

106. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7
m<Y = 6x +20
m<Y = 6(7) + 20
m<Y =
m<Y = 62 degrees

107. Example #12: Find x, m<X, m<Y
<X and <Y are complementary angles
m<X = 3x + 7
m<Y = 6x +20
m<X = 3x +7
m<X = 3(7) + 7
m<X =
m<X = 28 degrees
m<X + m<Y = 90
3x x +20 = 90
9x + 27 = 90
9x = 63
x = 7
m<Y = 6x +20
m<Y = 6(7) + 20
m<Y =
m<Y = 62 degrees