Question

how many sides does a regular polygon have if each interior angle measures $172^{\circ}$?

Explanation:

Step1: Relate interior to exterior angle

The interior angle and exterior angle of a regular polygon are supplementary, so:
$$180^\circ - 172^\circ = 8^\circ$$
This is the measure of one exterior angle.

Step2: Use total exterior angle sum

The sum of all exterior angles of any polygon is $360^\circ$. Let $n$ = number of sides.
$$n = \frac{360^\circ}{\text{Measure of one exterior angle}}$$
$$n = \frac{360}{8}$$

Step3: Calculate the number of sides

$$n = 45$$

Answer:

45