Given that lines $m \parallel n$, the angles formed by the transversal are related. The $36^\circ$ angle and the $x^\circ$ angle are alternate interior angles, which are equal when two parallel lines are cut by a transversal.

Therefore, we have: $x = 36^\circ$

So, the value of $x$ is $36^\circ$.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the sum of the interior angles on the same side of a transversal when it intersects two parallel lines?
2. What are corresponding angles, and how are they related when a transversal cuts through parallel lines?
3. How can you prove that two lines are parallel if given specific angle measures?
4. What is the relationship between alternate exterior angles when two lines are parallel?
5. How do you find the value of $x$ if the two lines were not parallel?

Tip: When working with parallel lines and a transversal, remember the key angle relationships: alternate interior angles, corresponding angles, and consecutive interior angles (supplementary).