The image shows three parallel lines $l$, $m$, and $n$, intersected by two transversal lines. The question asks to find the value of $x^\circ$, where $x$ represents an unknown angle formed by one of the transversals. The angle $135^\circ$ is given in the diagram between the lower transversal and the line $n$.

Steps to Solve:

1. Since $l \parallel m \parallel n$, all the corresponding and alternate angles formed by the transversal lines will be congruent.
2. The given $135^\circ$ angle and the unknown $x^\circ$ form a pair of alternate interior angles because both are on opposite sides of the transversal and between parallel lines $m$ and $n$.

According to the property of alternate interior angles: $x^\circ = 135^\circ$

Thus, the value of $x$ is: $x = 135^\circ$

Would you like any further details on this solution?

Here are 5 related questions for further exploration:

1. How do you find the value of angles using the properties of parallel lines and a transversal?
2. What is the difference between corresponding angles and alternate interior angles?
3. How can we determine if two lines are parallel based on the angles formed by a transversal?
4. What are supplementary angles, and how are they related to parallel lines?
5. How would the value of $x$ change if the angle given was $145^\circ$ instead?

Tip: When solving problems involving parallel lines, remember that alternate interior angles are always congruent, and corresponding angles also share this property.