To solve for $x$ and $y$ in the given diagram, let's analyze the geometry step by step.

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Step 1: Identify angle relationships

1. Parallel lines $m$ and $n$:
   The two parallel lines $m$ and $n$ are intersected by the diagonal transversal, forming alternate interior angles and supplementary angles.

2. Given angle $141^\circ$:

   • $141^\circ$ is an exterior angle on the upper side of line $m$.
   • Its corresponding interior angle on the opposite side of the transversal will be supplementary to $141^\circ$.
     Using the rule of supplementary angles: $\text{Interior angle} = 180^\circ - 141^\circ = 39^\circ$

   So, the angle on the opposite side of $141^\circ$ is $39^\circ$. This corresponds to $x^\circ$, because it is the alternate interior angle.

Thus, $x = 39^\circ$.

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Step 2: Solve for $y$

• Looking at the parallel line $n$ and its intersection with the transversal:
  • $y^\circ$ forms a linear pair with $x^\circ$ (they are adjacent angles on the same straight line).
  • Linear pairs are supplementary, so: $y = 180^\circ - x = 180^\circ - 39^\circ = 141^\circ$

Thus, $y = 141^\circ$.

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Final Answers:

$x = 39^\circ, \quad y = 141^\circ$

Let me know if you'd like further clarification!

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Related Questions:

1. What are alternate interior angles, and how are they used in parallel line problems?
2. Can you explain the concept of supplementary angles in more detail?
3. What happens if the transversal intersects two non-parallel lines instead?
4. How can you verify that lines $m$ and $n$ are parallel based on the given angles?
5. What is the sum of all angles formed around a point where a transversal intersects a parallel line?

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Tip: Always check if lines are parallel before applying angle rules such as alternate interior angles or supplementary relationships!