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Solving Angle Measures in Geometry

Learn how to find unknown angles using algebraic equations in geometry problems, such as calculating angles in triangles and right angles.

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Solving Angle Measures in Geometry

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  1. Findm∠ JKM. ALGEBRA 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 = 2 75 –5 = 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. x= 25 STEP1 Write and solve an equation to find the value of x. (5x – 10)° = 40° + 3x° 2x = 50 for Examples 3 and 4 GUIDED PRACTICE SOLUTION Apply the Exterior Angle Theorem. Solve for x.

  5. STEP2 Substitute 25 for xin 5x–10 to find 1. 5x–10 = 115 5 25 –10 = 180 1 + (5x – 10)° = 180° 1 + 115° = 1 = 65° ANSWER So measure of∠ 1in the diagram is 65°. for Examples 3 and 4 GUIDED PRACTICE

  6. 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

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

  8. 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 for Examples 3 and 4 GUIDED PRACTICE 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. x° +2x = 90° ACDis linear pair toACD. x = 30 So the measures of the acute angles are 30° and 2(30°) = 60° So 30° + ACD = 180°. ANSWER Therefore = ACD = 150°. for Examples 3 and 4 GUIDED PRACTICE Use the corollary to set up and solve an equation. Corollary to the Triangle Sum Theorem Solve for x.

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