Question

The $2$ nonadjacent sides of a regular pentagon ($5$ congruent and $5$ congruent interior angles) are extended to meet at point $X$ as shown in the above figure. Find the measure of $\angle X$.

• A. ${18}^{\circ }$
• B. ${30}^{\circ }$
• C. ${36}^{\circ }$
• D. ${45}^{\circ }$
• E. ${72}^{\circ }$

Answer 1

The correct option is C ${36}^{\circ }$

The sum of interior angles of the pentagon is $(n-2)\times {180}_{n=5}^0=3\times {180}^0=540$.
Since, it is a regular pentagon, the value of each of the interior angle is $\frac{540}{5}={108}^0$.

The two base angle of the triangle form linear pair with the interior angles of the pentagon.

Hence the measure of the two base angle of the triangle is ${180}^0-{108}^0={72}^0$.
Hence $x={180}^0-({72}^0+{72}^0)={180}^0-{144}^0={36}^0$.