To find the value of $x$, we need to apply the properties of parallel lines and angles.

Given that lines $m \parallel n$, the angles formed by the transversal are either corresponding, alternate interior, or consecutive interior angles.

In the diagram, the angle $113^\circ$ and $x^\circ$ are on opposite sides of the transversal and are supplementary because they form a linear pair (angles on the same straight line sum up to $180^\circ$).

Calculation:

$x + 113^\circ = 180^\circ$ $x = 180^\circ - 113^\circ = 67^\circ$

Thus, the value of $x$ is $\boxed{67^\circ}$.

Do you have any further questions or want additional details? Here are five related questions for further exploration:

1. What are alternate interior angles and how do they relate to parallel lines?
2. How can we use corresponding angles to solve problems involving parallel lines?
3. What other angle relationships exist when two lines are crossed by a transversal?
4. How can you prove that two lines are parallel using angle measurements?
5. How do supplementary and complementary angles differ?

Tip: Always remember that angles on a straight line sum to 180°, and this is often useful when dealing with parallel lines and transversals.