EXAMPLE 1 Identify complements and supplements SOLUTION In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair.

EXAMPLE 1 Identify complements and supplements

SOLUTION

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Because 122° + 58° = 180°, CAD and RST are supplementary angles.

Because BAC and CAD share a common vertex and side, they are adjacent.

Because 32°+ 58° = 90°, BAC and RST are complementary angles.

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

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

EXAMPLE 2 Find measures of a complement and a supplement

SOLUTION

a. Given that 1 is a complement of 2 and m 1 = 68°, find m 2.

m 2 = 90° – m 1 = 90° – 68° = 22°

a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.

EXAMPLE 2 Find measures of a complement and a supplement

b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°

SOLUTION

b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3.

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.

SOLUTION

EXAMPLE 3 Find angle measures

STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°.

Write equation.

(4x + 8)° + (x + 2)° = 180° Substitute.

5x + 10 = 180 Combine like terms.

5x = 170

x = 34

Subtract 10 from each side.

Divide each side by 5.

m BCE + m ∠ ECD = 180°

EXAMPLE 3 Find angle measures

STEP 2

Evaluate: the original expressions when x = 34.

m BCE = (4x + 8)° = (4 34 + 8)° = 144°

m ECD = (x + 2)° = ( 34 + 2)° = 36°

The angle measures are 144° and 36°.ANSWER

SOLUTION

GUIDED PRACTICE for Examples 2 and 3

3. Given that 1 is a complement of 2 and m 2 = 8o, find m 1.

82oANSWER

4. Given that 3 is a supplement of 4 and m 3 = 117o, find m 4.

63oANSWER

5. 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

SOLUTION

EXAMPLE 4 Identify angle pairs

To find vertical angles, look or angles formed by intersecting lines.

To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.

Identify all of the linear pairs and all of the vertical angles in the figure at the right.

1 and 5 are vertical angles.ANSWER

1 and 4 are a linear pair. 4 and 5 are also a linear pair.

ANSWER

SOLUTION

EXAMPLE 5 Find angle measures in a linear pair

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.

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.

ALGEBRA

EXAMPLE 5 Find angle measures in a linear pair

xo + 5xo = 180o

6x = 180

x = 30o

Write an equation.

Combine like terms.

Divide each side by 6.

The measures of the angles are 30o and 5(30)o = 150o.

ANSWER

GUIDED PRACTICE For Examples 4 and 5

ANSWER

No, no adjacent angles have their noncommon sides as opposite rays, 1 and 4 , 2 and 5, 3 and 6, these pairs of angles have sides that from two pairs of opposite rays.

Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain.

6.

GUIDED PRACTICE

7. The measure of an angle is twice the measure of its complement. Find the measure of each angle.

ANSWER 60°, 30°

For Examples 4 and 5