Question

how many sides does a regular polygon have if each interior angle measures 168°?

Explanation:

Step1: Find the measure of each exterior angle

The sum of an interior angle and its corresponding exterior angle in a polygon is \(180^\circ\). Given the interior angle is \(168^\circ\), so the exterior angle \(e = 180^\circ - 168^\circ= 12^\circ\).

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

The sum of the exterior angles of any polygon is always \(360^\circ\). Let \(n\) be the number of sides of the regular polygon. Then, since each exterior angle is \(e\) and there are \(n\) exterior angles, we have \(n\times e=360^\circ\). Substituting \(e = 12^\circ\) into the formula, we get \(n=\frac{360^\circ}{e}=\frac{360^\circ}{12^\circ}\).

Step3: Calculate the number of sides

\(\frac{360}{12} = 30\). So the number of sides \(n = 30\).

Answer:

The regular polygon has 30 sides.