We are given the following information in the diagram and question:

• The three friends are positioned at points $X$, $Y$, and $Z$.
• $\angle ZXY = 92^\circ$
• $\angle ZXY = 45^\circ$

To find the value of $\angle XZY$, we use the fact that the sum of the interior angles of a triangle must equal $180^\circ$.

We can write the equation for the sum of angles in triangle $XYZ$ as: $\angle ZXY + \angle ZYX + \angle XZY = 180^\circ$ Substitute the known values: $92^\circ + 45^\circ + \angle XZY = 180^\circ$ $137^\circ + \angle XZY = 180^\circ$ Solving for $\angle XZY$: $\angle XZY = 180^\circ - 137^\circ = 43^\circ$

Thus, the measure of $\angle XZY$ is 43°, and the correct answer is A.

Would you like further details on the solution? Here are some follow-up questions to explore this concept further:

1. How can we prove that the sum of angles in a triangle always equals 180°?
2. What other types of angle problems rely on the sum of interior angles?
3. How does this problem relate to the concept of supplementary angles?
4. Can we use this same method for quadrilaterals or other polygons?
5. How do angle bisectors affect the measures of angles in a triangle?

Tip: Always check if the angles provided form a triangle and verify their sum equals 180° to avoid miscalculations.