1. 1.6 Angle Pair Relationships
Geometry
1.6
Angle Pair Relationships
Essential Question: How can you use angle pairs to find angle measures?

2. Geometry 1.6 Angle Pair Relationships
Topics/Objectives
Know what Vertical Angles are.
Identify Linear Pairs.
Solve problems with Complementary Angles.
Solve problems with Supplementary Angles.
To define adjacent angles.
July 21, 2018
Geometry 1.6 Angle Pair Relationships

3. Another way to name angles…
Sometimes, for clarity and convenience, we will use a single number inside the angle to name it.
This is 1. 1
July 21, 2018
Geometry 1.6 Angle Pair Relationships

4. Geometry 1.6 Angle Pair Relationships
More than one angle 1 2 3
July 21, 2018
Geometry 1.6 Angle Pair Relationships

5. Geometry 1.6 Angle Pair Relationships
Vertical Angles
Two angles are vertical angles if their sides form two pairs of opposite rays. 3 1 2 4
1 & 2 are vertical angles.
3 & 4 are vertical angles.
July 21, 2018
Geometry 1.6 Angle Pair Relationships

6. Vertical Angles Property
Vertical Angles are congruent. ?
60°
60°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

7. Geometry 1.6 Angle Pair Relationships
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
Noncommon sides
July 21, 2018
Geometry 1.6 Angle Pair Relationships

8. Geometry 1.6 Angle Pair Relationships
Linear Pair Property
The sum of the angles of a linear pair is 180°.
70°
110° ?
July 21, 2018
Geometry 1.6 Angle Pair Relationships

9. Geometry 1.6 Angle Pair Relationships
Example 1 2 3 1 4 5
a. Are 1 and 2 a linear pair?
Yes no
b. Are 4 and 5 a linear pair? no
c. Are 3 and 5 vertical angles?
Yes
d. Are 1 and 3 vertical angles?
July 21, 2018
Geometry 1.6 Angle Pair Relationships

10. Geometry 1.6 Angle Pair Relationships
Example 2
These angles form a linear pair. The sum is 180°.
Find the measure of the three angles.
130° 2
These are vertical angles, and congruent.
50° 1
50°
These angles are vertical angles.
Vertical angles are congruent. 3
130°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

11. Geometry 1.6 Angle Pair Relationships
Example 3 page 1 A B
Solve for x, then find the measure of each angle.
(4x + 30)° E
(6x – 10)° C D
AEB and BEC form a linear pair.
What do we know about the sum of the angles of a linear pair?
The sum is 180°.
July 21, 2018
Geometry 1.6 Angle Pair Relationships

12. Geometry 1.6 Angle Pair Relationships
Example 3 page 2 A
Linear pair AEB and BEC means:
(4x + 30) + (6x – 10) = 180
10x + 20 = 180
10x = 160
x = 16
94° B
(4x + 30)° E
(6x – 10)°
86°
86°
94° C D
Then mAEB = 4(16) + 30 = 94
and mBEC = 6(16) – 10 = 86
July 21, 2018
Geometry 1.6 Angle Pair Relationships

13. Geometry 1.6 Angle Pair Relationships
Your Turn C
145° 1 2 3
(5x + 30)°
(2x – 4)° A B
1. Find the measure of 1, 2, 3.
2. Find the measure of ABC.
July 21, 2018
Geometry 1.6 Angle Pair Relationships

14. Geometry 1.6 Angle Pair Relationships
Solutions
180° C
145°
35°
(5x + 30)° 1
(2x – 4)°
35° 3 2 A B
145°
5x x – 4 = 180
7x + 26 = 180
7x = 154
x = 22
mABC = 5(22) + 30
= 140°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

15. Geometry 1.6 Angle Pair Relationships
Complementary Angles
Two angles are complementary if their sum is 90°.
These angles are complementary and adjacent.
65°
25°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

16. Geometry 1.4 Angles and Their Measure
Adjacent Angles
Adjacent angles have the same vertex, O, and one side in common, OB. They share no interior points. A B C O
There are THREE angles:
You cannot use the label O, since it would be unclear which angle that is.
AOB or BOA
BOC or COB
AOC or COA
July 21, 2018
Geometry 1.4 Angles and Their Measure

17. RST and VST are NOT adjacent angles.
July 21, 2018
Geometry 1.4 Angles and Their Measure

18. Geometry 1.6 Angle Pair Relationships
Complementary Angles
Two angles are complementary if their sum is 90°.
These angles are complementary and nonadjacent.
30°
60°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

19. Geometry 1.6 Angle Pair Relationships
Supplementary Angles
Angles are supplementary if their sum is 180°.
The angles are adjacent and supplementary (and a linear pair).
70°
110°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

20. Geometry 1.6 Angle Pair Relationships
Supplementary Angles
Angles are supplementary if their sum is 180°.
The angles are nonadjacent and supplementary.
80°
100°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

21. Geometry 1.6 Angle Pair Relationships
Example
5(40) – 50 = 150°
(5y – 50)°
Solve for y, then find m1.
30° 1
(4y – 10)°
150°
Vertical angles are congruent, so:
5y – 50 = 4y – 10
y =40
1 forms a linear pair with either of the 150° angles, so 1 is 30°.
July 21, 2018
Geometry 1.6 Angle Pair Relationships

22. Geometry 1.6 Angle Pair Relationships
Example
Find the measure of each angle.
(4x + 5)°
(3x + 8)°
4x x + 8 = 90
7x + 13 = 90
7x = 77
x = 11
49°
41°
This is a right angle, the angles are complementary. Their sum is 90°.
4(11) + 5 = 49°
3(11) + 8 = 41°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

23. Geometry 1.6 Angle Pair Relationships
Example 4
Find the value of each variable and the measure of each labeled angle.
130°
50°
(3x + 8)°
(5x – 20)°
50°
(5x + 4y)°
130°
3x + 8 = 5x – 20
-2x = -28
x = 14
5x + 4y = 130
5(14) + 4y = 130
70 + 4y = 130
4y = 60
y = 15
3(14) + 8 = 50°
July 21, 2018
Geometry 1.6 Angle Pair Relationships

24. Geometry 1.6 Angle Pair Relationships
Solve for x:
(6x + 10)
(4x + 40)
July 21, 2018
Geometry 1.6 Angle Pair Relationships

25. Geometry 1.6 Angle Pair Relationships
Solve for x:
(5x + 5)
(12x – 12)
July 21, 2018
Geometry 1.6 Angle Pair Relationships

26. Geometry 1.6 Angle Pair Relationships
Solve for x:
(x + 8)
(7x + 2)
July 21, 2018
Geometry 1.6 Angle Pair Relationships

27. Geometry 1.6 Angle Pair Relationships
Summarize
July 21, 2018
Geometry 1.6 Angle Pair Relationships