15. In the diagram shown, $P Q=S Q=Q R$ and $\angle S P Q=2 \times \angle R S Q$. What is the size of angle $Q R S$ ?
A $20^{\circ}$
B $25^{\circ}$
C $30^{\circ}$
D $35^{\circ}$
E $40^{\circ}$
15. $\mathbf{C}$ Let $\angle Q R S=x^{\circ}$. Then, as $S Q=Q R, \angle R S Q=x^{\circ}$ also. Now $\angle S P Q=2 \times \angle R S Q=2 x^{\circ}$. In triangle $P Q S, P Q=S Q$, so $\angle P S Q=\angle S P Q=2 x^{\circ}$. Therefore $\angle P S R=\angle P S Q+\angle R S Q=2 x^{\circ}+x^{\circ}=3 x^{\circ}$. The sum of the interior angles of a triangle is $180^{\circ}$. So, considering triangle $P S R$, $\angle S P R+\angle P S R+\angle P R S=180^{\circ}$. Therefore $2 x^{\circ}+3 x^{\circ}+x^{\circ}=180^{\circ}$.
So $6 x=180$, that is $x=30$.