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8.2

8.2. ESSENTIAL QUESTION: How do you c alculate the measures of interior and exterior angles of polygons?. Theorem 8.1 Polygon Interior Angles Theorem. The sum of the measures of the interior angles of a convex polygon with n sides is. Example 1.

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8.2

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  1. 8.2 ESSENTIAL QUESTION: How do you calculate the measures of interior and exterior angles of polygons?

  2. Theorem 8.1 Polygon Interior Angles Theorem The sum of the measures of the interior angles of a convex polygon with n sides is

  3. Example 1 Find the sum of the measures of the interior angles of a convex heptagon. Use the Polygon Interior Angles Theorem SOLUTION A heptagon has 7 sides. Use the Polygon Interior Angles Theorem and substitute 7 for n. (n– 2) · 180° = (7– 2) · 180° Substitute 7 for n. = 5 · 180° Simplify. = 900° Multiply. The sum of the measures of the interior angles of a convex heptagon is 900°. ANSWER

  4. Example 2 136° + 136° + 88° + 142° + 105° + mA =720° The sum is 720°. 607° + mA =720° Simplify. mA =113° Subtract 607°. Find the Measure of an Interior Angle Find the measure of A in the diagram. SOLUTION The polygon has 6 sides, so the sum of the measures of the interior angles is: (n– 2) · 180° = (6– 2) · 180° = 4 · 180° = 720°. Add the measures of the interior angles and set the sum equal to 720°. The measure of Ais113°. ANSWER

  5. Example 3 Interior Angles of a Regular Polygon Find the measure of an interior angle of a regular octagon. SOLUTION The sum of the measures of the interior angles of any octagon is: (n– 2) · 180° = (8– 2) · 180° = 6 · 180° = 1080°. Because the octagon is regular, each angle has the same measure. So, divide 1080° by 8 to find the measure of one interior angle. 1080° = 135° 8 The measure of an interior angle of a regular octagon is135°. ANSWER

  6. Checkpoint Find Measures of Interior Angles In Exercises 1–3, find the measure of G. 1. 2. 3. 4. Find the measure of an interior angle of a regular polygon with twelve sides.

  7. Checkpoint Find Measures of Interior Angles In Exercises 1–3, find the measure of G. 1. 72° ANSWER 2. 66° ANSWER 3. 130° ANSWER 4. Find the measure of an interior angle of a regular polygon with twelve sides. 150° ANSWER

  8. Theorem 8.2 Polygon Exterior Angles Theorem • The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°.

  9. Example 4 Find the Measure of an Exterior Angle Find the value of x. SOLUTION Using the Polygon Exterior Angles Theorem, set the sum of the measures of the exterior angles equal to 360°. 95° + 85° + 2x° + x°=360° Polygon Exterior Angles Theorem 180 + 3x=360 Combine like terms. 3x=180 Subtract 180 from each side. x=60 Divide each side by 3. The value of xis60. ANSWER

  10. Checkpoint Find Exterior Angle Measures Find the value of x. 5. 91 ANSWER 6. 21 ANSWER 7. 125 ANSWER 8. 30 ANSWER

  11. Worksheet 8.2B • Quiz Tomorrow Section 8.1 & 8.2

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