EXAMPLE 3 Find an angle measure

SOLUTION

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem.(2x – 5)° =70° + x°

Solve for x.x = 75

STEP 2Substitute 75 for x in 2x – 5 to find m∠JKM.

2x – 5 = 2 75 – 5 = 145

ALGEBRA Find m∠JKM.

The measure of ∠JKM is 145°.ANSWER

EXAMPLE 4 Find angle measures from a verbal description

ARCHITECTURE

The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle.

SOLUTION

First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

EXAMPLE 4 Find angle measures from a verbal description

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theoremx° + 2x° = 90°

Solve for x.x = 30

So, the measures of the acute angles are 30° and 2(30°) = 60° .

ANSWER

GUIDED PRACTICE for Examples 3 and 4

SOLUTION

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem. (5x – 10)° = 40° + 3x°

Solve for x.2x =50

Find the measure of 1 in the diagram shown.3.

x=25

GUIDED PRACTICE for Examples 3 and 4

STEP 2 Substitute 25 for x in 5x – 10 to find 1.

5x – 10 = 5 25– 10 = 115

1 + (5x – 10)° = 180

1 + 115° = 180°

1 = 65°

So measure of ∠1 in the diagram is 65°.ANSWER

GUIDED PRACTICE for Examples 3 and 4

SOLUTION

A + B + C = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

B = 2x = 2(30) = 60°

C = 3x = 3(30) = 90°

x

2x 3x

4. Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°.°

GUIDED PRACTICE for Examples 3 and 4

5. Find the measures of the acute angles of the right triangle in the diagram shown.

SOLUTION

Use the corollary to set up & solve an equation.

Corollary to the Triangle Sum Theorem(x – 6)° + 2x° = 90°

3x = 96

Solve for x.x = 32

Substitute 32 for x in equation x – 6 = 32 – 6 = 26°.

So, the measure of acute angle 2(32) = 64°ANSWER

GUIDED PRACTICE for Examples 3 and 4

6. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg?

A

B C Q

2x

xSOLUTION

First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

GUIDED PRACTICE for Examples 3 and 4

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theoremx° + 2x = 90°

Solve for x.x = 30

So the measures of the acute angles are 30° and 2(30°) = 60°

ACD is linear pair to ACD.

So 30° + ACD = 180°.

Therefore = ACD = 150°.ANSWER