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Finding Angle Measures Using Equations and Theorems

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In this guide, we'll explore how to determine angle measures through equations and geometric theorems. Utilizing principles such as the Exterior Angle Theorem and the Corollary to the Triangle Sum Theorem, we will solve for unknown angles based on given conditions. We'll tackle examples involving right triangles and provide step-by-step solutions to find acute angles and other relevant measures. Understanding these concepts is crucial in fields such as architecture and geometry.

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Finding Angle Measures Using Equations and Theorems

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  1. Findm∠ JKM. STEP1 Write and solve an equation to find the value of x. (2x – 5)° = 70°+ x° x = 75 STEP2 Substitute 75 for xin 2x–5 to find m∠ JKM. 2x–5 = 75 –5 = 2 145 ANSWER The measure of∠ JKMis145°. EXAMPLE 3 Find an angle measure SOLUTION Apply the Exterior Angle Theorem. Solve for x.

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

  4. Find the measure of 1 in the diagram shown. The measure of∠ 1in the diagram is 65°. ANSWER for Examples 3 and 4 GUIDED PRACTICE

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

  6. Find the measures of the acute angles of the right triangle in the diagram shown. ANSWER 26° and 64° for Examples 3 and 4 GUIDED PRACTICE

  7. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? A 2x x Q B C ANSWER ACD =150°. for Examples 3 and 4 GUIDED PRACTICE

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