## Geometry: Common Core (15th Edition)

Published by Prentice Hall

# Chapter 6 - Polygons and Quadrilaterals - 6-1 The Polygon Angle-Sum Theorems - Practice and Problem-Solving Exercises - Page 357: 30

#### Answer

This polygon has $10$ sides. m interior $\angle = 144^{\circ}$

#### Work Step by Step

If we have the measure of one exterior angle, we can find the number of sides this regular polygon has using the following formula. We'll use $n$ as the number of sides in this polygon: $36 = \frac{360}{n}$ Multiply each side of the equation by $n$ to get rid of the fraction: $36n = 360$ Divide each side by $36$ to solve for $n$: $n = 10$ This polygon has $10$ sides. To find the measure of an interior angle, we can use the corollary to the polygon angle-sum theorem, which states the following formula: m interior $\angle = \frac{(n - 2)(180)}{n}$ We know that $n$, the number of sides, for this polygon is $10$, so let's plug this piece of information in: m interior $\angle = \frac{(10 - 2)(180)}{10}$ Evaluate parentheses first: m interior $\angle = \frac{8(180)}{10}$ Multiply to simplify: m interior $\angle = \frac{1440}{10}$ Divide by $10$ to solve: m interior $\angle = 144^{\circ}$

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