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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.

If VY = XY = 75 and m∠VWY = 36°, what is m∠XWY?

This geometry problem involves solving for the angle m∠XWY in a triangle where VY = XY = 75 and m∠VWY = 36°. Key concepts include triangle angle sum and properties of isosceles triangles. Suitable for Grades 8-10.

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