In this diagram, we have two intersecting parallel lines with labeled angles. The goal is to solve for $x$.

The angles $(3x + 24)^\circ$ and $(5x - 12)^\circ$ are on alternate sides of the transversal line $AB$ and are equal because they are alternate interior angles, a property of parallel lines cut by a transversal.

Setting up the Equation

Since the angles are equal: $3x + 24 = 5x - 12$

Solving for $x$

1. Rearrange to isolate $x$: $24 + 12 = 5x - 3x$ $36 = 2x$

2. Divide by 2: $x = 18$

Conclusion

The value of $x$ is 18.

Would you like more details, or do you have any further questions?

Here are some related questions you might find helpful:

1. What are alternate interior angles?
2. How can we verify that two lines are parallel?
3. What is the difference between corresponding and alternate angles?
4. What are supplementary angles?
5. Can this method be used for non-parallel lines?

Tip: In diagrams with parallel lines, alternate interior angles formed by a transversal are always equal. This property is crucial for solving many geometric problems.