The given problem shows two parallel lines, $m \parallel n$, and a transversal crossing both lines, forming two angle expressions: $(5x - 14)^\circ$ and $(4x + 4)^\circ$.

To find the value of $x$, we can use the property that alternate interior angles are equal when two parallel lines are intersected by a transversal. These two angles are alternate interior angles.

Set the two expressions for the angles equal to each other:

$5x - 14 = 4x + 4$

Now, solve for $x$:

1. Subtract $4x$ from both sides: $x - 14 = 4$

2. Add 14 to both sides: $x = 18$

Thus, the value of $x$ is $18$.

Would you like a more detailed explanation or have further questions on the topic?

Here are five related questions you could explore:

1. How do alternate interior angles work with parallel lines and a transversal?
2. What are the other types of angles formed by a transversal intersecting parallel lines?
3. How can we use properties of parallel lines in geometric proofs?
4. What are vertical angles and how are they related to the problem?
5. What other angle relationships can we explore in geometric figures with parallel lines?

Tip: Always check if angles formed by a transversal with parallel lines are alternate interior, corresponding, or consecutive interior angles, as they each have unique properties.