1. 1.6 Describing Pairs of Angles
Geometry
1.6 Describing Pairs of Angles

2. 1.6 Describing Pairs of Angles
June 1, 2019

3. 1.6 Essential Question
How can you describe complementary and supplementary angles and use these descriptions to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
June 1, 2019

4. What You Will Learn Identify complementary and supplementary angles.
Identify linear pairs and vertical angles.
1.6 Describing Pairs of Angles
June 1, 2019

5. Recall from Last Lesson
Sometimes, for clarity and convenience, we will use a single number inside the angle to name it.
This is 1. 1
1.6 Describing Pairs of Angles
June 1, 2019

6. More than One Angle 1 2 3
1.6 Describing Pairs of Angles
June 1, 2019

7. Adjacent Angles
Adjacent angles have the same vertex, O, and one side in common, OB. They share no interior points. A B C
There are THREE angles: O
AOB or BOA
You cannot use the label O, since it would be unclear which angle that is.
BOC or COB
AOC or COA
1.6 Describing Pairs of Angles
June 1, 2019

8. RST and VST are NOT adjacent angles.
1.6 Describing Pairs of Angles
June 1, 2019

9. These angles are complementary AND adjacent.
Complementary Angles
Two angles are complementary if their sum is 90°.
These angles are complementary AND adjacent.
65°
25°
1.6 Describing Pairs of Angles
June 1, 2019

10. Complementary Angles Two angles are complementary if their sum is 90°.
These angles are complementary AND NONADJACENT.
Explain why:
30°
60°
1.6 Describing Pairs of Angles
June 1, 2019

11. Supplementary Angles Angles are supplementary if their sum is 180°.
These angles are adjacent AND supplementary (and a linear pair).
70°
110°
1.6 Describing Pairs of Angles
June 1, 2019

12. Supplementary Angles Angles are supplementary if their sum is 180°.
The angles are nonadjacent and supplementary.
Explain why:
80°
100°
1.6 Describing Pairs of Angles
June 1, 2019

13. Example 1
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.
1.6 Describing Pairs of Angles
June 1, 2019

14. Example 2
1.6 Describing Pairs of Angles
June 1, 2019

15. Linear Pair
Two adjacent angles are a linear pair if their noncommon sides are opposite rays.
Common Side
1 & 2 are a linear pair. 1 2 A B C
Noncommon sides
1.6 Describing Pairs of Angles
June 1, 2019

16. Linear Pair Property The sum of the angles of a linear pair is 180°.
110° ?
70°
1.6 Describing Pairs of Angles
June 1, 2019

17. Vertical Angles
Two angles are vertical angles if their sides form two pairs of opposite rays. 3 1 2
1 & 2 are vertical angles. 4
3 & 4 are vertical angles.
1.6 Describing Pairs of Angles
June 1, 2019

18. Vertical Angles Property
Vertical Angles are congruent. ?
60°
60°
1.6 Describing Pairs of Angles
June 1, 2019

19. Example 3 a. Are 1 and 2 a linear pair? Yes 1 2 3 4 5
b. Are 4 and 5 a linear pair? No
c. Are 3 and 5 vertical angles? No
d. Are 1 and 3 vertical angles?
Yes
1.6 Describing Pairs of Angles
June 1, 2019

20. Find the measure of the three angles.
Example 4
Find the measure of the three angles.
These angles form a linear pair. The sum is 180°.
130° 2
These are vertical angles, and congruent.
50° 1
50° 3
130°
These angles are vertical angles. Vertical angles are congruent.
1.6 Describing Pairs of Angles
June 1, 2019

21. Example 5 A Solve for x, then find the measure of each angle. B
AEB and BEC form a linear pair. C D
What do we know about the sum of the angles of a linear pair?
The sum is 180°.
1.6 Describing Pairs of Angles
June 1, 2019

22. Example 5 A 94° B Linear pair AEB and BEC means:
(4x + 30) + (6x – 10) = 180
10x + 20 = 180
10x = 160
x = 16
(4x + 30)° E
(6x – 10)°
86°
86°
94° C D
Then AEB = 4(16) + 30 = and BEC = 6(16) – 10 = 86
1.6 Describing Pairs of Angles
June 1, 2019

23. Your Turn Work through these two problems. C 145° 1 2 3 (5x + 30)°A B
1. Find the measure of 1, 2, 3.
2. Find the measure of ABC.
1.6 Describing Pairs of Angles
June 1, 2019

24. Your Turn Solutions 180° C (5x + 30)° (2x – 4)° 145° 35° A B 1 35° 3 2
mABC = 5(22) + 30
= 140°
1.6 Describing Pairs of Angles
June 1, 2019

25. Essential Question
How can you describe complementary and supplementary angles and use these descriptions to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
June 1, 2019

26. Assignment
1.6 Describing Pairs of Angles
June 1, 2019

27. 1.6 Describing Pairs of Angles
Day 2

28. Essential Question
When two lines intersect, how do you know if two angles are congruent or supplementary and how do you use this information to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
June 1, 2019

29. Vertical Angles are congruent.
Quick Review 4 1 2 3
Vertical Angles are congruent.
1  2 & 3  4
1.6 Describing Pairs of Angles
June 1, 2019

30. The angles of a Linear Pair are Supplementary
Quick Review 4 1 2 3
The angles of a Linear Pair are Supplementary
m1 + m4 = 180
m4 + m2 = 180
1.6 Describing Pairs of Angles
June 1, 2019

31. Quick Review Two angles are supplementary if their sum is 180.
Two angles are complementary if their sum is 90.
1.6 Describing Pairs of Angles
June 1, 2019

32. Example 6 Solve for x, then find the angle measures. Solution:
AEB and DEA are a linear pair. The sum of the angles in a linear pair is 180°.
6x + (3x + 45) = 180
9x = 135
x = 15 B
6(15) = 90° A
6x° E C
(3x + 45)°
3(15) = 90° D
1.6 Describing Pairs of Angles
June 1, 2019

33. Example 7 Solve for y, then find m1.
Vertical angles are congruent, so:
5y – 50 = 4y – 10
y = 40
5(40) – 50 = 150°
(5y – 50)°
30° 1
(4y – 10)°
150°
1 forms a linear pair with either of the 150° angles, so 1 is 30°.
1.6 Describing Pairs of Angles
June 1, 2019

34. Example 8 Find the measure of each angle. 4x + 5 + 3x + 8 = 90
49°
41°
This is a right angle, the angles are complementary. Their sum is 90°.
4(11) + 5 = 49°
3(11) + 8 = 41°
1.6 Describing Pairs of Angles
June 1, 2019

35. Example 9
Find the value of each variable and the measure of each labeled angle.
5x + 4y = 130
5(14) + 4y = 130
70 + 4y = 130
4y = 60
y = 15
130°
50°
(3x + 8)°
(5x – 20)°
50°
(5x + 4y)°
130°
3x + 8 = 5x – 20
-2x = -28
x = 14
3(14) + 8 = 50°
1.6 Describing Pairs of Angles
June 1, 2019

36. Five Sample Problems For You To Do 1.6 Describing Pairs of Angles
June 1, 2019

37. 1. Solve for x. (4x + 40) (6x + 10) 1.6 Describing Pairs of Angles
June 1, 2019

38. 2. Solve for x. (12x – 12) (5x + 5) 1.6 Describing Pairs of Angles
June 1, 2019

39. 3. Solve for x. (x + 8) (7x + 2) 1.6 Describing Pairs of Angles
June 1, 2019

40. 4. Solve for x & y. (7x + 4) (9y + 3) (5y  5) (13x + 16)
1.6 Describing Pairs of Angles
June 1, 2019

41. 5. Solve for x. A is supplementary to B. mA = (2x + 10)
mB = (3x  5)
2x x  5 = 180
5x + 5 = 180
5x = 175
x = 35
1.6 Describing Pairs of Angles
June 1, 2019

42. Essential Question
When two lines intersect, how do you know if two angles are congruent or supplementary and how do you use this information to find angle measures?
1.4 Perimeter and Area in the Coordinate Plane
June 1, 2019

43. Assignment
1.6 Describing Pairs of Angles
June 1, 2019