Answer
This polygon has $5$ sides.
The measure of an interior angle is $108^{\circ}$.
Work Step by Step
If we have the measure of one exterior angle, we can find the number of sides this polygon has using the following formula. We'll use $n$ as the number of sides in this polygon:
$72 = \frac{360}{n}$
Multiply each side of the equation by $x$ to get rid of the fraction:
$72n = 360$
Divide each side by $72$ to solve for $n$:
$n = 5$
This polygon has $5$ 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 $5$, so let's plug this piece of information in:
m interior $\angle = \frac{(5 - 2)(180)}{5}$
Evaluate parentheses first:
m interior $\angle = \frac{3(180)}{5}$
Multiply to simplify:
m interior $\angle = \frac{540}{5}$
Divide by $5$ to solve:
m interior $\angle = 108$
