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Examples on how to find the measures of interior angles in regular and non-regular polygons using the interior angle sum theorem. It also covers the calculation of exterior angles in a convex regular nonagon.
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ARCHITECTURE The Pentagon in Washington, D.C. is shaped like a regular pentagon. Find the sum of the measures of the interior angles of the largest pentagon-shaped section of the Pentagon building. Since the Pentagon is a convex polygon, we can use the Interior Angle Sum Theorem. S = 180( n - 2) Interior Angle Sum Theorem = 180(5 - 2) n = 5 = 180(3) or 540 Simplify.
The sum of the measures of the interior angles is 540.
The measure of an interior angle of a regular polygon is 135. Find the number of sides of the polygon. Use the Interior Angle Sum Theorem to write an equation to solve for n , the number of sides. S = 180( n - 2) Interior Angle Sum Theorem (135) n = 180( n - 2) S = 135 n 135 n = 180 n - 360 Distributive Property 360 = 45 n Subtract 135 n and add 360 to each side. 8 = n Divide each side by 45. The polygon has 8 sides.
ALGEBRA Find the measure of each interior angle. Since n = 4, the sum of the measures of the interior angles is 180(4 – 2) or 360. Write an equation to express the sum of the measures of the interior angles of the polygon.
360 = m R + m S + m T + m U Sum of measures of angles 360 = x + 2 x + 3 x + 4 x Substitution 360 = 10 x Combine like terms. 36 = x Divide each side by 10. Use the value of x to find the measure of each angle. m R = 36, m S = 2 ∙ 36 or 72, m T = 3 ∙ 36 or 108, m U = 4 ∙ 36 or 144.
Example 4 Exterior Angles Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHI****. At each vertex, extend a side to form one exterior angle. The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles. 9 n = 360 n = measure of each exterior angle n = 40 Divide each side by 9.
The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.