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 $\angle = 144^{\circ}$
