Question

problem solving the interior angles of a regular polygon each measure 165°. how many sides does the polygon have? the polygon has □ sides.

Explanation:

Step1: Find the measure of an exterior angle

The sum of an interior angle and its corresponding exterior angle of a polygon is \(180^{\circ}\). Given the interior angle is \(165^{\circ}\), then the exterior angle \(e = 180 - 165=15^{\circ}\)

Step2: Use the formula for the sum of exterior angles of a polygon

The sum of the exterior angles of any polygon is \(360^{\circ}\). Let \(n\) be the number of sides of the polygon. Using the formula \(n=\frac{360}{e}\) (where \(e\) is the measure of one exterior angle)

Substitute \(e = 15^{\circ}\) into the formula: \(n=\frac{360}{15}\)

Step3: Calculate the value of \(n\)

\(n = 24\)

Answer:

\(24\)