3. The diagram shows two congruent regular pentagons and a triangle. The angles marked $x^{\circ}$ are equal.
What is the value of $x$ ?
A 24
B 30
C 36
D 40
E 45
SolUtion
C
Let $P, Q$ and $R$ be the points shown in the diagram.
The interior angles of a regular pentagon are all $108^{\circ}$.
Because the two pentagons are congruent, $P R=P Q$.
Therefore $\angle P R Q=\angle P Q R=x^{\circ}$.
Because the sum of the angles in a triangle is $180^{\circ}$, from the triangle $P Q R$, we have $x^{\circ}+x^{\circ}+\angle Q P R=$ $180^{\circ}$. Therefore $\angle Q P R=180^{\circ}-2 x^{\circ}$.
The sum of the angles at a point is $360^{\circ}$. Therefore from the angles at the point $P$, we have
$$
\left(180^{\circ}-2 x^{\circ}\right)+108^{\circ}+x^{\circ}+108^{\circ}=360^{\circ} \text {. }
$$

That is
$$
396^{\circ}-x^{\circ}=360^{\circ} .
$$

It follows that
$$
x^{\circ}=396^{\circ}-360^{\circ}=36^{\circ} \text {. }
$$