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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!

Learn how to solve for x in a geometry problem involving parallel lines cut by a transversal. In this problem, same-side interior angles are supplementary, and the solution involves basic angle relationships.

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