The problem shows two parallel lines, $m \parallel n$, with a transversal intersecting them. The given angles at the intersections are $(4x + 8)^\circ$ and $(4x - 20)^\circ$.

Because the lines $m$ and $n$ are parallel and the transversal creates corresponding angles, these angles are congruent (equal).

Thus, we can set the two angle expressions equal to each other:

$4x + 8 = 4x - 20$

Now let's solve for $x$.

1. Subtract $4x$ from both sides: $8 = -20$ This gives a contradiction, which means that the angles might not be corresponding but rather supplementary (since their sum must equal $180^\circ$).

Thus, the correct relation should be:

$(4x + 8) + (4x - 20) = 180$

Now let's solve this equation:

$4x + 8 + 4x - 20 = 180$ $8x - 12 = 180$ $8x = 192$ $x = \frac{192}{8} = 24$

Thus, the value of $x$ is $24$.

Would you like more details on any of the steps? Here are five related questions to deepen understanding:

1. What are corresponding angles in parallel line geometry?
2. How can we identify supplementary angles in different transversal cases?
3. How can we confirm whether angles should be set equal or supplementary?
4. What would happen if the transversal intersected non-parallel lines?
5. How can parallel line properties help solve problems with more complex angles?

Tip: Always check whether the angles formed by a transversal on parallel lines are corresponding, alternate, or supplementary, as this guides the correct equation to solve.