To find the measures of angles $x$, $y$, and $z$ in the figure, we can use the properties of parallel lines. Since $m \parallel n$, we know that corresponding and alternate interior angles are congruent.

1. Given: Angle $\angle x = 49^\circ$ (as indicated in the image).

2. Finding $\angle z$:

   • Since $m \parallel n$ and $\angle x$ is on the top line, $\angle z$ is an alternate interior angle with $\angle x$.
   • Therefore, $\angle z = 49^\circ$.

3. Finding $\angle y$:

   • $\angle y$ and $\angle z$ are on a straight line (they are supplementary angles).
   • Thus, $\angle y + \angle z = 180^\circ$.
   • Since $\angle z = 49^\circ$, we have: $\angle y = 180^\circ - 49^\circ = 131^\circ$

Final Answers:

• $\angle x = 49^\circ$
• $\angle z = 49^\circ$
• $\angle y = 131^\circ$

Would you like further details on each step?

1. What are the properties of alternate interior angles?
2. How can we find the measure of an angle if we know its supplementary angle?
3. Why do parallel lines allow us to use corresponding and alternate angle relationships?
4. What are supplementary angles, and how do they relate to straight lines?
5. How would the solution change if angle $x$ were a different value?

Tip: When working with parallel lines and a transversal, remember that corresponding and alternate interior angles are congruent.