12. In the diagram, $P Q R S$ is a square, $P S T$ is an equilateral triangle and $S R U V W$ is a regular pentagon.
What is the size of angle WTS?
A $35^{\circ}$
B $36^{\circ}$
C $37^{\circ}$
D $38^{\circ}$
E $39^{\circ}$
12. E The four angles at point $S$ include an interior angle of each of an equilateral triangle, a square and a regular pentagon. These are $60^{\circ}, 90^{\circ}$ and $108^{\circ}$ respectively. As the sum of the angles which meet at a point is $360^{\circ}, \angle T S W$ is $(360-(60+90+108))^{\circ}=102^{\circ}$.
The equilateral triangle and the square have side $P S$ in common and the square and the regular pentagon have side $R S$ in common, so the triangle, square and pentagon have equal side-lengths. Therefore $S W=S T$ and hence triangle $S W T$ is isosceles.
So $\angle W T S=\angle T W S=\frac{1}{2}(180-102)^{\circ}=39^{\circ}$.