22. In trapezium $P Q R S$, the sides $P Q$ and $S R$ are parallel. Angle $R S P$ is $120^{\circ}$ and $P S=S R=\frac{1}{3} P Q$. What is the size of angle $P Q R$ ?
A $15^{\circ}$
B $22.5^{\circ}$
C $25^{\circ}$
D $30^{\circ}$
E $45^{\circ}$
The diagram shows the trapezium described with points $X$ and $Y$ added on $P Q$ so that $P X=X Y=Y Q=P S=S R$. Since angle $R S P=120^{\circ}$, angle $S P Q=60^{\circ}$ using co-interior angles and so triangle $S P X$ is equilateral. Similarly, it can easily be shown that triangles $S X R$ and $R X Y$ are also equilateral. In triangle $R Y Q$ we then have $R Y=Y Q$ and angle $R Y Q=120^{\circ}$ using angles on a straight line adding to $180^{\circ}$. Hence triangle $R Y Q$ is isosceles and so angle $Y Q R=\frac{1}{2}\left(180^{\circ}-120^{\circ}\right)=30^{\circ}$. Therefore angle $P Q R$ is $30^{\circ}$.