Answer
$\angle PXY = 45^{\circ}$
Work Step by Step
If $P$ is the incenter, that means this is the point where all angle bisectors meet.
Because $\angle YXZ$ is bisected by $\overline{PX}$, $m \angle PXY$ is congruent to $m \angle PXZ$ and would be half of $m \angle YXZ$, which is $90^{\circ}$. Both angles, $\angle PXY$ and $\angle PXZ$, have a measure of $45^{\circ}$.
