The measure of the indicated angles are:
m∠PXY = 45°
m∠PYZ = 40°
m∠PZY = 25°.
The incenter of a triangle is the point where the bisectors of the interior angles intersect. It is equidistant from the three sides of the triangle. This implies that:
m∠PXY = m∠PXZ
m∠PYX = m∠PYZ
m∠PZX = m∠PZY
The angle m∠X = 90° so;
m∠PXY = 90/2
m∠PXY = 45°
Since the angle m∠PYX is equal to 20° then;
m∠PYZ = 20° + 20°
m∠PYZ = 40°
Considering the right triangle YXZ, angle Z is derived as;
Z + 90° + 40° = 180° {sum of interior angles of a triangle}
Z + 130° = 180°
Z = 180° - 130°
Z = 50°
so m∠PZY = 50°/2
m∠PZY = 25°
Therefore, the measures of the indicated angles are; m∠PXY = 45°, m∠PYZ = 40° and m∠PZY = 25°