To solve this problem, we need to determine the value of $x$ that makes line $m$ parallel to line $n$. When two lines are parallel, corresponding angles or alternate interior angles are equal.

In the figure:

• $m \angle 2 = (20x + 52)^\circ$
• $m \angle 7 = (40x - 48)^\circ$

Angles $\angle 2$ and $\angle 7$ are alternate interior angles. For $m$ to be parallel to $n$, these angles must be equal. Therefore, we can set up the following equation:

$20x + 52 = 40x - 48$

Solving for $x$:

1. Subtract $20x$ from both sides: $52 = 20x - 48$

2. Add $48$ to both sides: $100 = 20x$

3. Divide both sides by $20$: $x = 5$

Thus, the value of $x$ that makes line $m$ parallel to line $n$ is 5.

The correct answer is: (A) 5

Would you like further details or have any questions?

Here are some related questions:

1. What if we wanted to prove lines are perpendicular instead of parallel?
2. How do alternate interior angles work with non-parallel lines?
3. Can we use the same approach for lines intersecting at angles other than right angles?
4. How would the solution change if $m$ and $n$ were skew lines in 3D space?
5. What if the angles were given as expressions with different variables?

Tip: When working with parallel lines, remember that alternate interior angles are equal, which helps in setting up equations.