In the image, we are given two parallel lines $m \parallel n$ and a transversal that creates angles $x^\circ$, $y^\circ$, and $49^\circ$. We are tasked with finding the values of $x$ and $y$.

Solution:

1. Corresponding Angles Theorem: Since $m \parallel n$, we can use the fact that corresponding angles are equal. The angle marked as $49^\circ$ is corresponding to angle $y^\circ$. Therefore, $y = 49^\circ$

2. Alternate Interior Angles Theorem: We can use alternate interior angles to find $x$. Angle $x^\circ$ and $49^\circ$ are alternate interior angles because they are on opposite sides of the transversal and inside the parallel lines $m$ and $n$. Therefore, $x = 49^\circ$

Thus, both $x$ and $y$ are $49^\circ$.

Final Answers:

• $x = 49^\circ$
• $y = 49^\circ$

Would you like further details or clarification on this? Here are some related questions to deepen your understanding:

1. What are alternate interior angles and how are they formed?
2. How can you distinguish between corresponding and alternate angles?
3. If one of the angles formed by the transversal is $90^\circ$, what would be the measure of the other angles?
4. Can the sum of corresponding angles ever be more or less than $180^\circ$?
5. How do we use the parallel lines property to determine angles in more complex configurations?

Tip: Always look for parallel lines and transversals in geometry problems—they often lead to equal angle relationships like corresponding and alternate interior angles.