1. Angles PA

2. Measurement of an Angle

3. Here are some common angles and their measurements.
To denote the measure of an angle we write an “m” in front of the symbol for the angle.
Here are some common angles and their measurements. 1 2 4 3

4. Congruent Angles
So, two angles are congruent if and only if they have the same measure.
So, The angles below are congruent.
Means Congruent
Means Equal

5. Types of Angles
An acute angle is an angle that measures less than 90 degrees.
A right angle is an angle that measures exactly 90 degrees.
An obtuse angle is an angle that measures more than 90 degrees.
acute
right
obtuse

6. Types of Angles
A straight angle is an angle that measures 180 degrees. (It is the same as a line.)
When drawing a right angle we often mark its opening as in the picture below.
right angle
straight angle

7. Perpendicular Lines
Two lines are perpendicular if they intersect to form a right angle. See the diagram.
Suppose angle 2 is the right angle. Then since angles 1 and 2 are supplementary, angle 1 is a right angle too. Similarly, angles 3 and 4 are right angles.
So, perpendicular lines intersect to form four right angles. 2 1 3 4

8. Perpendicular Lines The symbol for perpendicularity is
So, if lines m and n are perpendicular, then we write m n

9. Adjacent Angles
Adjacent angles share a common vertex and one common side.
Adjacent angles are “side by side” and share a common ray.
Common
Ray
15º
45º
Common
Vertex

10. These are examples of adjacent angles.
45º
80º
35º
55º
130º
50º
85º
20º

11. These angles are NOT adjacent.
Adjacent Angles
These angles are NOT adjacent.
100º
50º
35º
35º
55º
45º

12. When 2 lines intersect, they make vertical angles.
Two angles formed by intersecting lines and have no sides in common but share a common vertex.
Are congruent.
Common Vertex
75º
When 2 lines intersect, they make vertical angles.
105º
105º
75º

13. Vertical angles are opposite one another.
75º
105º
105º
75º

14. Vertical angles are opposite one another.
75º
105º
105º
75º

15. Vertical angles are congruent (equal).
150º
30º
150º
30º

16. Name the Vertical Angles
Two angles that are opposite angles. Vertical angles are congruent.
Name the Vertical Angles
 1   4
 5   8, 1 2
 6   7
 2   3 3 4 5 6 7 8

17. Supplementary Angles Add up to 180º.
40º
120º
60º
140º
Adjacent and Supplementary Angles
Supplementary Angles but not Adjacent

18. Supplementary Angles
Two angles are supplementary if their measures add up to
If two angles are supplementary each angle is the supplement of the other.
If two adjacent angles together form a straight angle as below, then they are supplementary. 1 2

19. Complementary Angles Add up to 90º.
20º
20º
70º
70º
Complementary Angles but not Adjacent
Adjacent and Complementary Angles

20. Complementary Angles
Two angles are if their measures add upcomplementary to
If two angles are complementary, then each angle is called the complement of the other.
If two adjacent angles together form a right angle as below, then they are complementary. A B C 1 2

21. Supplementary vs. Complementary How do I remember?
The way I remember is this:
C comes before S in the alphabet.
90 comes before 180 when I count.
Complementary is 90, Supplementary is 180.

22. Guess Who?
I am an angle.

23. Guess Who?
I am an angle.
I have 180°

24. Guess Who?
I am an angle.
I have 180°
I look like this:

25. Guess Who? I am an angle. I have 180° I look like this: Supplementary
Complementary

26. Guess Who?
I am two adjacent angles.

27. Guess Who? I am two adjacent angles.
I look like an “L” with a line in the middle.

28. Guess Who? I am two adjacent angles.
I look like an “L” with a line in the middle.
I add up to 90°
I look like this:

29. Guess Who? I am two adjacent angles.
I look like an “L” with a line in the middle.
I add up to 90°
I look like this:
Complementary
Supplementary

30. Guess Who?
Complementary
Supplementary

31. Guess Who?
Complementary
Supplementary

32. Review
Complementary angles are…….

33. Review
Complementary angles are…….

34. Review
Supplementary Angles are…..

35. Practice Time!

36. Find the missing angle I know that these angles are complementary.
They must add up to 90°
So……
90 – 55 = 35
The missing angle is 35 55 x

37. You try. Are they supplementary or complementary?
Find the missing side. x 20

38. You try. Are they supplementary or complementary? complementary
Find the missing side.
90 – 20 = 70
The missing angle
Is 70 x 20

39. One More 50 x

40. Find the missing angle 120 x I know these are supplementary angles.
Supplementary angles add up to 180.
The given angle is So…..
180 – 120 = 60
The missing angle is 60

41. Find the missing angle 130 x What kind of angles?
What’s the missing angle?

42. Find the missing angle 130 x What kind of angles? Supplementary
What’s the missing angle?
Adds up to 180, so…..
180 – 130 = 50

43. Find the missing angle x 30
Do this one on your own.

44. Directions: Identify each pair of angles as adjacent, vertical, supplementary, complementary, or none of the above.

45. #1
120º
60º

46. #1
120º
60º
Supplementary Angles
Adjacent Angles

47. #2
60º
30º

48. #2
60º
30º
Complementary Angles

49. #3
75º
75º

50. #3
Vertical Angles
75º
75º

51. #4
60º
40º

52. #4
60º
40º
None of the above

53. #5
60º
60º

54. #5
60º
60º
Vertical Angles

55. #6
135º
45º

56. #6
135º
45º
Supplementary Angles
Adjacent Angles

57. #7
25º
65º

58. #7
25º
65º
Complementary Angles
Adjacent Angles

59. #8
90º
50º

60. #8
90º
50º
None of the above

61. Directions: Determine the missing angle.

62. #1 ?º
45º

63. #1
135º
45º

64. #2 ?º
65º

65. #2
25º
65º

66. #3 ?º
35º

67. #3
35º
35º

68. #4 ?º
50º

69. #4
130º
50º

70. #5 ?º
140º

71. #5
140º
140º

72. #6 ?º
40º

73. #6
50º
40º

74. Transversal
Definition: A line that intersects two or more lines in a plane at different points is called a transversal.
When a transversal t intersects line n and m, eight angles of the following types are formed:
Exterior angles
Interior angles
Consecutive interior angles
Alternative exterior angles
Alternative interior angles
Corresponding angles t m n

75. Corresponding Angles
Corresponding Angles: Two angles that occupy corresponding positions.
The corresponding angles are the ones at the same location at each intersection
 2   6
 3   7
 4   8
 1   5 1 2 3 4 5 6 7 8

76. Angles and Parallel Lines
If two parallel lines are cut by a transversal, then the following pairs of angles are congruent.
Corresponding angles
Alternate interior angles
Alternate exterior angles

77. Proving Lines Parallel
If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

78. Lesson 2-4: Angles and Parallel Lines
Alternate Angles
Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair).
Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal.
 3   6,  4   5
 2   7,  1   8 1 2 3 4 5 6 7 8
Lesson 2-4: Angles and Parallel Lines

79. Lesson 2-4: Angles and Parallel Lines
Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers. t 16 15 14 13 12 11 10 9 8 7 6 5 3 4 2 1 s D C B A
m<2=80° m<3=100° m<4=80°
m<5=100° m<6=80° m<7=100° m<8=80°
m<9=100° m<10=80° m<11=100° m<12=80°
m<13=100° m<14=80° m<15=100° m<16=80°
Lesson 2-4: Angles and Parallel Lines

80. Proving Lines Parallel
If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

81. Ways to Prove Two Lines Parallel
Show that corresponding angles are equal.
Show that alternative interior angles are equal.
In a plane, show that the lines are perpendicular to the same line.

82. Homework
Pg 305 #6-14e, 18-32e (just answers) Pg 309 #6-24e (just answers)