1. Write and solve an equation to find the value of x.
EXAMPLE 3
Find an angle measure
ALGEBRA
Find m∠ JKM.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
(2x – 5)° =
70° + x°
Apply the Exterior Angle Theorem.
x = 75
Solve for x.
STEP 2
Substitute 75 for x in 2x – 5 to find m∠ JKM.
2x – 5 = 2 75
– 5 =
145
The measure of ∠ JKM is 145°.
ANSWER

2. 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.

3. Find angle measures from a verbal description
EXAMPLE 4
Find angle measures from a verbal description
Use the corollary to set up and solve an equation.
x° + 2x° =
90°
Corollary to the Triangle Sum Theorem
x = 30
Solve for x.
So, the measures of the acute angles are 30° and 2(30°)
= 60° .
ANSWER

4. Find the measure of 1 in the diagram shown.
GUIDED PRACTICE
for Examples 3 and 4
Find the measure of 1 in the diagram shown.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
(5x – 10)° =
40° + 3x°
Apply the Exterior Angle Theorem.
2x = 50
Solve for x. x= 25

5. GUIDED PRACTICE
for Examples 3 and 4
STEP 2
Substitute 25 for x in 5x – 10 to find
5x – 10 = 5 25
– 10 =
115
1 + (5x – 10)° =
180
° =
180°
1 =
65°
So measure of ∠ 1 in the diagram is 65°.
ANSWER

6. GUIDED PRACTICE
for Examples 3 and 4 x 2x 3x
Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°.
SOLUTION
A + B C =
180°
x + 2x + 3x =
180°
6x =
180°
x =
30°
B =
2x =
2(30) =
60°
C =
3x =
3(30) =
90°

7. Use the corollary to set up & solve an equation.
GUIDED PRACTICE
for Examples 3 and 4
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.
(x – 6)° + 2x° =
90°
Corollary to the Triangle Sum Theorem
3x = 96
x = 32
Solve for x.
Substitute 32 for x in equation x – 6 =
32 – 6 =
26°.
So, the measure of acute angle 2(32) =
64°
ANSWER

8. GUIDED PRACTICE for Examples 3 and 4
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 x
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.

9. Use the corollary to set up and solve an equation.
GUIDED PRACTICE
for Examples 3 and 4
Use the corollary to set up and solve an equation.
x° + 2x =
90°
Corollary to the Triangle Sum Theorem
x = 30
Solve for x.
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