The problem asks to find the measure of $\angle WXY$.

From the image, we can see that $\triangle WXY$ is formed by three angles:

• $\angle W = 61^\circ$
• $\angle Y = 62^\circ$
• $\angle X$ is the one we need to find.

We know that the sum of the angles in any triangle is always $180^\circ$. So, we can set up the equation:

$\angle W + \angle Y + \angle X = 180^\circ$

Substitute the known values:

$61^\circ + 62^\circ + \angle X = 180^\circ$

Simplify:

$123^\circ + \angle X = 180^\circ$

Solve for $\angle X$:

$\angle X = 180^\circ - 123^\circ = 57^\circ$

Thus, the measure of $\angle WXY$ is $\boxed{57^\circ}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we prove that the sum of interior angles in a triangle is always $180^\circ$?
2. How can we calculate an unknown angle if two angles of a triangle are known?
3. What is the significance of exterior angles in a triangle?
4. How would this process change for non-triangular polygons (e.g., quadrilaterals)?
5. How does the triangle inequality theorem relate to angles in triangles?

Tip: When solving for an unknown angle in a triangle, always remember that the sum of the angles in any triangle equals $180^\circ$.