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The problem asks to determine the value of \( x \) that would prove lines \( p \parallel q \) given that they are cut by the transversal \( m \). The angles formed by the transversal are provided as:

- \( 2x - 3^\circ \) on one side of the transversal.
- \( x + 9^\circ \) on the other side of the transversal.

### Step-by-Step Solution:

When two lines are parallel, alternate interior angles formed by a transversal are equal. Thus, for \( p \parallel q \), we need to set the two given angles equal to each other:

\[
2x - 3 = x + 9
\]

### Solving 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 proves \( p \parallel q \) is \( \boxed{12} \).

### Conclusion:
The correct answer is 12.

Would you like more details on this solution, or have any questions? Here are some related questions for further practice:

1. What other angle relationships could confirm parallel lines?
2. How would the solution change if one angle was exterior?
3. How do corresponding angles help prove parallelism?
4. How would you verify if \( m \) is not a transversal but another type of line?
5. Could \( x \) have other values if the angles were supplementary instead of equal?

**Tip:** Always check if the problem involves corresponding, alternate, or consecutive interior angles when dealing with transversals and parallel lines!

Lines p and q are cut by transversal m. What value of x would prove p || q? Given angles are (2x - 3)° and (x + 9)°.

Learn how to solve for x in a geometry problem where lines p and q are cut by a transversal. Using the alternate interior angles theorem, the equation (2x - 3)° = (x + 9)° is solved step-by-step to determine the value of x that proves p || q.

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