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# 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.$72^\circ$

Last updated date: 20th Jun 2024
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Hint: First, we will use that the sum of interior angles of the pentagon is $\left( {n - 2} \right) \times 180^\circ$, where $n$ is the number of sides. Then take $n = 5$ in this formula and then in a regular pentagon, the value of each of the interior angle is computed by dividing the above value of 5

We are given that the 2 non adjacent sides of a regular pentagon (5 congruent and 5 congruent interior angles) are extended to meet at point $X$.
We know that the sum of interior angles of the pentagon is $\left( {n - 2} \right) \times 180^\circ$, where $n$ is the number of sides.

Since we know that the sides of the given figure are $n = 5$.
Substituting the value of $n$ in the above formula of sum of interior angles of the pentagon, we get
$\Rightarrow \left( {5 - 2} \right) \times 180^\circ \\ \Rightarrow 3 \times 180^\circ \\ \Rightarrow 540^\circ \\$
Since we are given that a regular pentagon, the value of each of the interior angle is computed by dividing the above value of 5, we get
$\Rightarrow \dfrac{{540^\circ }}{5} \\ \Rightarrow 108^\circ \\$
We know that the two base angles of the triangle form a linear pair with the interior angles of the pentagon.
So, we will the measure of the two base angle of the triangle by subtracting $180^\circ$ by $108^\circ$, we get
$\Rightarrow 180^\circ - 108^\circ \\ \Rightarrow 72^\circ \\$
Thus we have from the above figure that
$\Rightarrow 180^\circ - \left( {72^\circ + 72^\circ } \right) \\ \Rightarrow 180^\circ - 144^\circ \\ \Rightarrow 36^\circ \\$

Hence, option C is correct.

Note: In solving this question, we have multiplied the given ratio with some unknown variable. Then use the fact that the sum of angle of the triangle is $180^\circ \left( {n - 2} \right)$, where $n$ is the number of sides to obtain a linear equation.