13. In trapezium $P Q R S, \angle R S P=2 \times \angle S P Q$ and $\angle S P Q=2 \times \angle P Q R$. Also $\angle Q R S=k \times \angle P Q R$. What is the value of $k$ ?
A 2
B 3
C 4
D 5
E 6
13. D Let the size in degrees of $\angle P Q R$ and of $\angle Q R S$ be $x$ and $k x$. Therefore the size of $\angle S P Q$ and of $\angle R S P$ are $2 x$ and $2 \times 2 x=4 x$ respectively. Since the angles between parallel lines (sometimes called co-interior or allied angles) add to $180^{\circ}$, we have $2 x+4 x=180$. This has solution $x=30$. Similarly $x+k x=180$ and hence $30 k=150$. Therefore the value of $k$ is 5 .