Given the information and the geometry of the problem, we have four points $W$, $X$, $Y$, and $Z$, forming some kind of polygon or triangle. You mentioned that angles $\angle W$ and $\angle Z$ are given as $52^\circ$ and $38^\circ$ respectively. We are asked to solve for $m\angle WXY$, which is the measure of angle $WXY$.

Based on the information, it seems like these points are vertices of a quadrilateral or triangle, but since angle $\angle WXY$ is mentioned, I will assume we are dealing with a triangle where $WXY$ is the angle to solve for.

Using the Triangle Angle Sum Theorem:

For any triangle, the sum of the interior angles is always $180^\circ$. If $\triangle WXY$ is formed, the angles inside the triangle must add up to $180^\circ$.

Given:

• $\angle W = 52^\circ$
• $\angle Z = 38^\circ$

Thus, $m\angle WXY = 180^\circ - (52^\circ + 38^\circ)$.

Let's compute that:

$m\angle WXY = 180^\circ - 90^\circ = 90^\circ$

So, the measure of $m\angle WXY$ is $90^\circ$.

Would you like any further details or explanations?

Here are 5 related questions for deeper understanding:

1. What is the Triangle Angle Sum Theorem?
2. How do we use exterior angles to find unknown angles in polygons?
3. Can angle sums be applied to other polygons, such as quadrilaterals?
4. How would the approach change if the shape was not a triangle?
5. What other ways can we calculate unknown angles in geometry?

Tip: Always double-check whether the given points form a triangle, quadrilateral, or other polygon before using specific angle-sum properties.