The image shows two parallel lines $p$ and $q$ cut by a transversal $m$. We are given two expressions for the angles on either side of the transversal: $(2x - 3)^\circ$ on the left and $(x + 9)^\circ$ on the right. The question asks for the value of $x$ that would prove $p \parallel q$.

Solution:

For $p \parallel q$, the corresponding angles formed by the transversal must be equal. Thus, the angles given by $(2x - 3)^\circ$ and $(x + 9)^\circ$ must be equal.

We can set up the following equation: $2x - 3 = x + 9$

Now, solve for $x$:

1. Subtract $x$ from both sides: $2x - x - 3 = 9$ This simplifies to: $x - 3 = 9$

2. Add 3 to both sides: $x = 12$

Thus, the value of $x$ that makes the lines $p \parallel q$ is $\boxed{12}$.

Answer: 12

Would you like more details on this or have any other questions?

Here are some related questions for further practice:

1. What is the corresponding angle theorem in geometry?
2. How do you solve for $x$ when two alternate interior angles are equal?
3. What conditions must be met for two lines to be parallel when cut by a transversal?
4. What is the relationship between corresponding and alternate angles?
5. How can you determine if two lines are parallel using only angle measures?

Tip: When working with parallel lines and transversals, remember that corresponding angles are always equal when the lines are parallel.