To solve this problem, we note that $m \parallel n$, which makes the lines parallel, and the angles formed by the transversal are either corresponding angles, alternate interior angles, or supplementary angles. Here's the reasoning:

1. The given angle $138^\circ$ and the unknown angle $x^\circ$ are supplementary angles because they are on a straight line formed by the transversal.
2. Therefore, the relationship between these two angles is: $138^\circ + x = 180^\circ$
3. Solving for $x$: $x = 180^\circ - 138^\circ$ $x = 42^\circ$

Thus, the value of $x$ is $\mathbf{42^\circ}$.

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Related Questions

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How do corresponding angles help in solving problems involving parallel lines?
3. Can you explain the sum of angles on a straight line in more depth?
4. What other properties of parallel lines and transversals are commonly tested in geometry?
5. How do you distinguish between interior, exterior, and adjacent angles in similar problems?

Tip: Always look for key geometric relationships like parallel lines, transversals, and angle pair properties when solving such problems.