Answer
$\angle XYZ = 40^{\circ}$
Work Step by Step
If $P$ is the incenter, that means this is the point where all angle bisectors meet.
Because $\angle XYZ$ is bisected by $\overline{PY}$, $\angle XYZ$ is the sum of $\angle XYP$ and $\angle ZYP$, so if $\angle XYP$ is $20^{\circ}$, then $\angle XYZ$ is $40^{\circ}$.
