19. In rectangle $P Q R S$, the ratio of $\angle P S Q$ to $\angle P Q S$ is $1: 5$. What is the size of $\angle Q S R$ ?
A $15^{\circ}$
B $18^{\circ}$
C $45^{\circ}$
D $72^{\circ}$
E $75^{\circ}$
Solution: $\mathbf{E}$
Let $\angle Q S R=x^{\circ}$. Since $P S$ is parallel to $Q R$, the alternate angles $\angle P Q S$ and $\angle Q S R$ are equal. So $\angle P Q S=x^{\circ}$. Therefore, as $\angle P S Q: \angle P Q S$ $=1: 5, \angle P S Q=\frac{1}{5} x^{\circ}$. Therefore, from the right angled triangle $P Q S$ we deduce that $x+\frac{1}{5} x+90=180$ and so $\frac{6}{5} x=90$. Therefore $x=\frac{5}{6} \times 90=75$. So $\angle Q S R$ is $75^{\circ}$.