In this problem, we have two parallel lines, $m$ and $n$, and a transversal line that intersects them, forming angles $x^\circ$, $y^\circ$, and a known angle of $125^\circ$.

Solution

Since $m \parallel n$, we can use the properties of parallel lines and the angles created by a transversal.

1. Finding $x$:

   • $x$ and $125^\circ$ are on the same side of the transversal and are supplementary (they form a linear pair on the line $n$).
   • Therefore: $x + 125 = 180^\circ$
   • Solving for $x$: $x = 180 - 125 = 55^\circ$

2. Finding $y$:

   • $y$ is an alternate interior angle to $125^\circ$ because it is on the opposite side of the transversal and inside the two parallel lines.
   • Alternate interior angles are equal when lines are parallel, so: $y = 125^\circ$

Final Answer

• $x = 55^\circ$
• $y = 125^\circ$

Would you like more details or have any questions about this solution?

Here are 5 related questions to consider:

1. What are alternate interior angles, and why are they equal when lines are parallel?
2. How do supplementary angles relate to parallel lines and transversals?
3. What is the difference between alternate interior and corresponding angles?
4. How can you determine angle measures if only given one angle on a transversal with parallel lines?
5. What other angle relationships occur with parallel lines and a transversal?

Tip: Always look for angle pairs (like alternate interior or corresponding angles) when working with parallel lines and a transversal—they often simplify the solution.