The problem involves two parallel lines $m$ and $n$, with a transversal cutting through them. The angles formed are given as $(3x - 23)^\circ$ and $(7x - 7)^\circ$.

Since $m \parallel n$, these two angles are corresponding angles, which means they must be equal.

Thus, we can set up the equation:

$3x - 23 = 7x - 7$

Solving for $x$:

1. Subtract $3x$ from both sides: $-23 = 4x - 7$

2. Add 7 to both sides: $-16 = 4x$

3. Divide by 4: $x = -4$

Thus, the value of $x$ is $-4$.

Would you like more details on how corresponding angles work or have any questions?

Here are some related questions for practice:

1. What are alternate interior angles, and how are they used in parallel line problems?
2. Can you explain the concept of supplementary angles when a transversal intersects parallel lines?
3. How would the problem change if the given angles were alternate exterior angles instead of corresponding angles?
4. If the lines were not parallel, could you still use the corresponding angle rule?
5. What other types of angles are formed when a transversal intersects two parallel lines?

Tip: In parallel line problems, look for angle relationships such as corresponding, alternate interior, or same-side interior angles, as they often lead to helpful equations.