Answer
$m \angle XZY = 25^{\circ}$
Work Step by Step
If $P$ is the incenter, that means this is the point where all angle bisectors meet.
Because $\angle XZY$ is bisected by $\overline{PZ}$, $\angle XZY$ is the sum of $\angle PZX$ and $\angle PZY$. So we need to find the measure of $\angle XZY$ to figure out the measure of $\angle PZX$.
We already know the measures of $\angle YXZ$, which is $90^{\circ}$, and $\angle XYZ$, which is $40^{\circ}$. We also know that the measures of the interior angles of a triangle add up to $90^{\circ}$, so if we subtract the sum of the two known angles from $180^{\circ}$, then we'll find the measure of $\angle XZY$. Let's set up the equation using this information:
$m \angle XZY = 180 - (m \angle YXZ + m \angle XYZ)$
Let's plug in our "knowns":
$m \angle XZY = 180 - (90 + 40)$
Evaluate what's in parentheses first, according to order of operations:
$m \angle XZY = 180 - (130)$
Subtract to solve:
$m \angle XZY = 50$
$m \angle PZX$ is half of $m \angle XZY$; therefore, $m \angle XZY$ is $25^{\circ}$.
