The image shows a geometry problem where you are asked to find the measure of angle $\angle XWY$, given that $VY = XY = 75$ and $m\angle VWY = 36^\circ$. The diagram indicates a triangle, with one right angle at $\angle WXY$.

Solution:

We can solve the problem using the following steps:

1. Since $VY = XY = 75$, triangle $VXY$ is isosceles.
2. In an isosceles triangle, the angles opposite the equal sides are equal. So, $\angle VXY = \angle XYV$.
3. The sum of the angles in triangle $VWY$ must be $180^\circ$. We are given $m\angle VWY = 36^\circ$, and we know that $\angle WXY = 90^\circ$ (since a right angle is marked at $X$).

Let’s calculate $m\angle XWY$:

• In triangle $VWY$, the sum of the interior angles is: $m\angle VWY + m\angle XWY + m\angle WXY = 180^\circ$ Substituting the known values: $36^\circ + m\angle XWY + 90^\circ = 180^\circ$ Simplifying: $m\angle XWY + 126^\circ = 180^\circ$ $m\angle XWY = 180^\circ - 126^\circ = 54^\circ$

Therefore, the measure of $\angle XWY$ is $\boxed{54^\circ}$.

Would you like more details on any part of the solution? Here are some related questions you might explore:

1. What are the properties of an isosceles triangle?
2. How do you calculate the angles in a triangle?
3. Why do the angles in any triangle always sum to $180^\circ$?
4. How does knowing a right angle helps simplify the problem?
5. What would change if the side lengths were different?

Tip: In geometry, identifying congruent angles and right angles often simplifies complex problems.