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In this image, two parallel lines \( m \parallel n \) are given with a transversal. The angles formed are \( (2x - 4)^\circ \) and \( (3x - 12)^\circ \), which are alternate interior angles. Since \( m \parallel n \), alternate interior angles are congruent. This gives us the equation:

\[
2x - 4 = 3x - 12
\]

### Solving for \( x \):
1. Subtract \( 2x \) from both sides:
   \[
   -4 = x - 12
   \]

2. Add 12 to both sides:
   \[
   8 = x
   \]

Thus, the value of \( x \) is \( \boxed{8} \).

Would you like a detailed breakdown or have any questions?

### Related questions:
1. What are alternate interior angles, and why are they congruent in this case?
2. How would the problem change if the lines weren't parallel?
3. How can we check if our solution for \( x \) is correct?
4. What happens if the angles are supplementary instead of congruent?
5. Can this method be used to solve for angles on other types of transversal intersections?

### Tip:
Always double-check which angle relationships are given—parallel lines often lead to congruent or supplementary angles depending on the type of intersection.

Learn how to solve for x using the Alternate Interior Angles Theorem in a problem involving parallel lines and a transversal. The problem provides the equation 2x - 4 = 3x - 12. Suitable for students in Grades 8-10.

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