In the image, we have two parallel lines $m \parallel n$, and two angles formed by a transversal cutting across these lines. The angles given are $(4x + 7)^\circ$ and $(5x - 19)^\circ$. Since the lines are parallel and the angles are on opposite sides of the transversal, they are alternate interior angles. Alternate interior angles are congruent (equal) when lines are parallel.

Thus, we set the angles equal to each other:

$4x + 7 = 5x - 19$

Solving for $x$:

1. Subtract $4x$ from both sides: $7 = x - 19$

2. Add 19 to both sides: $7 + 19 = x$ $x = 26$

Thus, the value of $x$ is 26.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How do you know when angles are congruent in parallel line problems?
2. What other angle relationships exist between parallel lines and a transversal?
3. Can you solve for $x$ if the lines weren't parallel?
4. What happens if one of the angles is expressed as an inequality?
5. How do you solve problems involving same-side interior angles?

Tip: In parallel line problems, always identify angle pairs such as corresponding, alternate interior, or alternate exterior angles. They often give key clues to solving the problem.