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Angles in Polygons: Calculating Interior and Exterior Angles, Study Guides, Projects, Research of Architecture

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.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

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Angles of Polygons
Example 1 Interior Angles of Regular Polygons
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.
Example 2 Sides of a Polygon
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 = 135n
135n = 180n - 360 Distributive Property
360 = 45n Subtract 135n and add 360 to each side.
8 = n Divide each side by 45.
The polygon has 8 sides.
Example 3 Interior Angles of Nonregular Polygons
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 + 2x + 3x + 4x Substitution
360 = 10x 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.
9n = 360 n = measure of each exterior angle
n = 40 Divide each side by 9.
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Angles of Polygons

Example 1 Interior Angles of Regular Polygons

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.

Example 2 Sides of a Polygon

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.

Example 3 Interior Angles of Nonregular Polygons

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.