1. 1. The sum of two numbers is 90 and one number is 4 times the other
1. The sum of two numbers is 90 and one number is 4 times the other. Write an equation and solve to find the numbers.
ANSWER
x + 4x = 90; 18, 72
2. Find m ABD. What kind of angle is it?
ANSWER
180° , straight

2. Measure Geometric Figures
Target
Measure Geometric Figures
You will…
Use special angle relationships to find angle measures.

3. Special Angle Pairs Vocabulary
adjacent angles – share a common vertex, a common side, and no common interior points
adjacent angle pairs
ABD and DBE
ABD and DBC
DBE and EBC
ABE and EBC
NOT adjacent angle pairs
ABD and EBC
no common side
ABE and DBC
common interior points

4. Special Angle Pairs Vocabulary
linear pair – two adjacent angles whose noncommon sides form opposite rays
linear pairs
ABD and DBC
ABE and EBC
3 and 4
4 and 6
3 and 5
5 and 6

5. Special Angle Pairs Vocabulary
vertical angles – two angles whose sides form two pairs of opposite rays
vertical angles
3 and 6
4 and 5
complementary angles – two angles whose measures total 90° (no other conditions are necessary)
supplementary angles – two angles whose measures total 180° (no other conditions are necessary)

6. EXAMPLE 1
Identify complements and supplements
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.
SOLUTION
Because 32°+ 58° = 90°, BAC and RST are complementary angles.
Because 122° + 58° = 180°, CAD and RST are supplementary angles.
Because BAC and CAD share a common vertex and side, they are adjacent.

7. GUIDED PRACTICE
for Example 1
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1.
FGK and GKL,
HGK and GKL,
FGK and HGK
ANSWER

8. GUIDED PRACTICE
for Example 1
Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain. 2.
No, they do not share a common vertex.
No, they have common interior points.
ANSWER

9. EXAMPLE 2
Find measures of a complement and a supplement
Given that 1 is a complement of and m = 68°,
find m
SOLUTION
a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.
m = 90° – m = 90° – 68° = 22°

10. EXAMPLE 2
Find measures of a complement and a supplement
b. Given that is a supplement of 4 and m = 56°,
find m 3.
SOLUTION
b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship.
m 3 = 180° – m 4 = 180° –56° = 124°

11. EXAMPLE 3
Find angle measures
Sports
When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

12. EXAMPLE 3 Find angle measures SOLUTION STEP 1
Use the fact that the sum of the measures of supplementary angles is 180°.
m BCE + m ∠ ECD = 180°
Write equation.
(4x + 8)° + (x + 2)° = 180°
Substitute.
5x + 10 = 180
Combine like terms.
5x = 170
Subtract 10 from each side.
x = 34
Divide each side by 5.

13. EXAMPLE 3
Find angle measures
SOLUTION
STEP 2
Evaluate: the original expressions when x = 34.
m BCE = (4x + 8)° = ( )° = 144°
m ECD = (x + 2)° = ( )° = 36°
The angle measures are 144° and 36°.
ANSWER

14. GUIDED PRACTICE
for Examples 2 and 3
3. Given that is a complement of and m = 8o, find m 1.
82o
ANSWER
4. Given that is a supplement of and m = 117o, find m
63o
ANSWER
LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)o and m PQR = (9x + 1)o.
ANSWER
26o, 64o

15. EXAMPLE 4
Identify angle pairs
Identify all of the linear pairs and all of the vertical angles in the figure at the right.
SOLUTION
To find vertical angles, look or angles formed by intersecting lines.
1 and are vertical angles.
ANSWER
To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.
1 and 4 are a linear pair and are also a linear pair.
ANSWER

16. Find angle measures in a linear pair
EXAMPLE 5
Find angle measures in a linear pair
Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each.
ALGEBRA
SOLUTION
Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.
xo + 5xo = 180o
Write an equation.
6x = 180
Combine like terms.
x = 30o
Divide each side by 6.
The measures of the angles are 30o and (30)o = 150o.
ANSWER

17. GUIDED PRACTICE
For Examples 4 and 5
Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain. 6.
ANSWER
No, no adjacent angles have their noncommon sides as opposite rays, and , and 5, and 6, these pairs of angles have sides that from two pairs of opposite rays.

18. GUIDED PRACTICE
For Examples 4 and 5
7. The measure of an angle is twice the measure of its complement. Find the measure of each angle.
ANSWER
60°, 30°