This problem involves parallel lines cut by a transversal, where we are asked to find the value of angle $x$.

Step-by-step solution:

1. Recognize the angle relationships:

   • The lines $l \parallel m \parallel n$, and the transversal crosses all three lines.
   • The angle marked as $135^\circ$ and the angle $x^\circ$ are same-side interior angles.

2. Same-side interior angles: When two parallel lines are cut by a transversal, the same-side interior angles are supplementary, meaning their sum is $180^\circ$.

3. Set up the equation: $x + 135^\circ = 180^\circ$

4. Solve for $x$: $x = 180^\circ - 135^\circ = 45^\circ$

Final Answer:

$x = 45^\circ$

Would you like more details or have any other questions?

Here are some related questions you might consider:

1. What are alternate interior angles, and how do they differ from same-side interior angles?
2. How would the problem change if $135^\circ$ were an exterior angle?
3. Can you identify other angles that are congruent to $x$ in this diagram?
4. What other types of angle pairs are formed by parallel lines and a transversal?
5. How would the problem change if the lines were not parallel?

Tip: Remember, same-side interior angles are always supplementary when the lines are parallel!