To find the value of $x$, let's analyze the angles in this parallel line and transversal setup.

1. Since lines $m$ and $n$ are parallel, the transversal creates pairs of corresponding angles and alternate interior angles.
2. The angle labeled $102^\circ$ and the angle labeled $x^\circ$ are corresponding angles because they are on the same side of the transversal and on corresponding positions relative to the parallel lines.
3. For corresponding angles, if two lines are parallel, the angles are equal.

Therefore: $x = 102^\circ$

So, the value of $x$ is: $x = 102$

Would you like a more detailed explanation of the properties of parallel lines and transversals?

Here are some additional related questions you might find helpful:

1. What are alternate interior angles and when are they equal?
2. How can you identify corresponding angles when two lines are parallel?
3. What other angle relationships exist when a transversal intersects parallel lines?
4. How would the solution change if $m$ and $n$ were not parallel?
5. What are the steps to prove two lines are parallel using angle relationships?

Tip: Remember, corresponding angles are equal when two parallel lines are intersected by a transversal.