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In this diagram, two lines \( l \) and \( m \) are intersected by a transversal. The angles formed on opposite sides of the transversal are given as:

- \( (9x + 25)^\circ \) on line \( l \),
- \( (13x - 19)^\circ \) on line \( m \).

These angles appear to be **corresponding angles** since they occupy the same relative position at the intersection points. For two corresponding angles to be equal, the lines \( l \) and \( m \) must be parallel.

Thus, we can set the angles equal to each other and solve for \( x \):

\[
9x + 25 = 13x - 19
\]

Let's solve this equation.

\[
9x + 25 = 13x - 19
\]
Subtract \( 9x \) from both sides:
\[
25 = 4x - 19
\]
Add 19 to both sides:
\[
44 = 4x
\]
Divide by 4:
\[
x = 11
\]

Now that we have \( x = 11 \), we can substitute this value back into the expressions for the angles to verify:

- \( (9x + 25) = 9(11) + 25 = 99 + 25 = 124^\circ \),
- \( (13x - 19) = 13(11) - 19 = 143 - 19 = 124^\circ \).

Since both angles are indeed equal, our solution is correct. Therefore, \( x = 11 \).

Would you like any further explanation or clarification? 

Here are 5 related questions:
1. What are alternate interior angles, and how do they relate to parallel lines?
2. How can you identify corresponding angles in a transversal setup?
3. What is the significance of the transversal line in geometry?
4. How do we know the lines \( l \) and \( m \) are parallel in this situation?
5. Could these angles be supplementary instead, and how would that change the problem?

**Tip:** When solving for variables in geometry, always check whether the angle relationships are equal (like corresponding angles) or supplementary (like consecutive interior angles).

Two angles formed by intersecting lines l and m, represented by the expressions (9x + 25)° and (13x - 19)°, are corresponding angles. Solve for x.

Learn how to solve a geometry problem involving corresponding angles formed by intersecting lines. The angles are expressed as (9x + 25)° and (13x - 19)°, and we find the value of x. This solution is suitable for students in Grades 7-10 and uses algebraic methods.

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