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4-2 Angles of Triangles You classified triangles by their side or angle measures. Apply the Triangle Angle-Sum Theorem. Apply the Exterior Angle Theorem.

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Presentation transcript:

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

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.

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

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

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 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°

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

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.

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