O is any point on the bisector of the acute angle ∠XYZ. The line OP∥ZY. Then, △YPO is an isosceles right-angled triangle.
Isosceles but not right angled
∠POY = ∠ZYO (alt. ∠s) = ∠PYO.
So, PY = PO.
Now, ∠XYZ < 90∘
⇒POY < 45∘ ⇒∠PYO < 45∘
∴∠YPO > 90∘
Hence, Δ YPO is an isosceles but not a right angled triangle.