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The image shows two parallel lines \( m \parallel n \) crossed by a transversal, forming alternate interior angles. The two angles given are:

- \( (10x - 13)^\circ \) 
- \( (9x + 1)^\circ \)

Since the lines are parallel, we know that alternate interior angles are congruent. Therefore, we can set the angles equal to each other:

\[
10x - 13 = 9x + 1
\]

### Solving for \( x \):

1. Subtract \( 9x \) from both sides:
\[
10x - 9x - 13 = 1
\]
\[
x - 13 = 1
\]

2. Add 13 to both sides:
\[
x = 14
\]

Thus, the value of \( x \) is \( \boxed{14} \).

Let me know if you want further details or if you have any questions!

Here are some related questions you might find useful:

1. How do we know that alternate interior angles are equal when lines are parallel?
2. What are the other types of angles formed by a transversal intersecting parallel lines?
3. How would the solution change if the lines were not parallel?
4. What is the difference between alternate interior angles and corresponding angles?
5. How can you verify that the value of \( x \) is correct?

**Tip**: Remember, alternate interior angles are only congruent when the lines are parallel!

Explore how to find the value of x when two parallel lines are intersected by a transversal using the Alternate Interior Angles Theorem. This detailed solution involves setting up and solving a simple algebraic equation derived from angle congruence.

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