13. $P Q R S T$ is a regular pentagon. The point $U$ lies on $S T$ such that $\angle Q P U$ is a right angle. What is the ratio of the interior angles in triangle $P U T$ ?
A $1: 3: 6$
B $1: 2: 4$
C $2: 3: 4$
D $1: 4: 8$
E $1: 3: 5$
SolUtion
A

The interior angles of a regular pentagon are all $108^{\circ}$. [You are asked to prove this in Problem 13.1.] Therefore $\angle P T U=\angle Q P T=108^{\circ}$.
Hence $\angle U P T=\angle Q P T-\angle Q P U=108^{\circ}-90^{\circ}=18^{\circ}$.
Because the angles in a triangle have sum $180^{\circ}$, it follows that $\angle T U P=180^{\circ}-108^{\circ}-18^{\circ}=54^{\circ}$.

Therefore the ratio of the interior angles in the triangle PUT is $18: 54: 108=1: 3: 6$.