Question

in the figure below, 2 nonadjacent sides of a regular pentagon (5 congruent sides and 5 congruent interior angles) are extended until they meet at point x. what is the measure of ∠x? 18° 30° 36° 45° 72°

Explanation:

Step1: Find interior - angle of pentagon

The formula for the measure of an interior angle of a regular polygon is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$, where $n = 5$ for a pentagon. So, $\theta=\frac{(5 - 2)\times180^{\circ}}{5}=\frac{3\times180^{\circ}}{5}=108^{\circ}$.

Step2: Find exterior - angle of pentagon

The exterior angle of a regular polygon is supplementary to the interior angle. Let the exterior angle be $\alpha$. Then $\alpha=180^{\circ}-\theta$. So, $\alpha = 180^{\circ}-108^{\circ}=72^{\circ}$.

Step3: Calculate $\angle X$

In the triangle formed by the two extended non - adjacent sides of the pentagon, the two base angles are the exterior angles of the pentagon. Using the angle - sum property of a triangle ($180^{\circ}$ for the sum of interior angles of a triangle), if the two base angles are each $72^{\circ}$, then $\angle X=180^{\circ}-2\times72^{\circ}=180^{\circ}-144^{\circ}=36^{\circ}$.

Answer:

C. 36°