This is a problem involving parallel lines and a transversal. The lines $m \parallel n$ are intersected by a transversal, creating several angles. Given that $m \parallel n$, the angles formed by the transversal have special relationships. Specifically:

• Corresponding angles are equal.
• Alternate interior angles are equal.
• Same-side interior angles are supplementary (sum to 180°).

From the image:

• You are given $y^\circ$ and $43^\circ$ as angles on line $m$, and $x^\circ$ on line $n$.

Step-by-Step Solution:

1. Angle $y$:

   • The angle $y^\circ$ is an alternate interior angle to the given $43^\circ$ angle (since $m \parallel n$ and the transversal intersects them).
   • Therefore, $y = 43^\circ$.

2. Angle $x$:

   • The angle $x^\circ$ and the $43^\circ$ angle are corresponding angles because they are on the same side of the transversal and between the parallel lines $m \parallel n$.
   • Thus, $x = 43^\circ$.

Final Answers:

• $x = 43^\circ$
• $y = 43^\circ$

Would you like more details or have any other questions?

Here are 5 questions to extend this concept:

1. What is the relationship between alternate interior angles and parallel lines?
2. How do you prove that corresponding angles are equal when lines are parallel?
3. What happens to the angle relationships if the lines are not parallel?
4. Can two angles be supplementary but not adjacent? Explain with an example.
5. What are vertical angles, and how do they relate to this scenario?

Tip: Always double-check if the given lines are parallel before applying angle theorems.