To solve for $x$ in this problem, we will use the concept of alternate interior angles. The diagram shows two parallel lines $m$ and $n$ cut by a transversal.

The angles on the opposite sides of the transversal but inside the parallel lines are equal because they are alternate interior angles.

From the diagram: $(6x - 9)^\circ = (3x + 10)^\circ$

Since these angles are equal: $6x - 9 = 3x + 10$

Now, solve for $x$:

1. Subtract $3x$ from both sides: $3x - 9 = 10$

2. Add 9 to both sides: $3x = 19$

3. Divide by 3: $x = \frac{19}{3} \approx 6.33$

Thus, $x \approx 6.33$.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the definition of alternate interior angles?
2. How do parallel lines affect the angles formed by a transversal?
3. Can you identify other pairs of equal angles in the diagram?
4. How would the solution change if the angles were supplementary instead of equal?
5. How can you verify your solution using substitution?

Tip: Always check the properties of angles formed by parallel lines and a transversal, as these relationships are key to solving many geometry problems.