The given problem involves two parallel lines $m$ and $n$, with a transversal cutting through them. We are asked to find the value of $x$ based on the angle expressions provided: $(4x - 8)^\circ$ and $(7x - 20)^\circ$.

Step-by-step Solution:

1. Since $m \parallel n$, and the angles formed by the transversal are corresponding angles, these angles must be equal.

   So, we can set up the equation: $(4x - 8) = (7x - 20)$

2. Now, solve for $x$: $4x - 8 = 7x - 20$ Subtract $4x$ from both sides: $-8 = 3x - 20$ Add 20 to both sides: $12 = 3x$ Divide both sides by 3: $x = 4$

Thus, the value of $x$ is $\boxed{4}$.

Would you like more details on this solution, or do you have any other questions?

Here are 5 related questions to practice:

1. What are alternate interior angles, and how can you identify them in parallel line problems?
2. If $m \parallel n$ and one angle is $5x - 15$ degrees and the corresponding angle is $6x - 25$, how would you solve for $x$?
3. How do you determine the value of angles formed by a transversal cutting through non-parallel lines?
4. What happens to the relationship between angles if the lines are not parallel?
5. Can you calculate the value of $x$ if two supplementary angles are given as $3x + 10$ and $5x - 20$?

Tip: In transversal problems, always look for equal corresponding angles, alternate interior angles, or supplementary angles to set up your equations.