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The image shows a geometry problem involving parallel lines, with a diagram at the bottom. The instruction states: "Find the value of \( x \) so that line \( l \parallel m \). State the converse used."

In the diagram, angles formed by a transversal cutting through two lines \( l \) and \( m \) are shown. The angles are labeled with expressions involving \( x \):
- One angle measures \( 16x^\circ \),
- Another angle measures \( 28x^\circ \),
- A given angle between the two parallel lines measures \( 48^\circ \).

### Steps to solve:
1. **Parallel Lines Condition**: For two lines \( l \) and \( m \) to be parallel, the angles formed by a transversal must follow certain angle relationships. One possible relation could be that corresponding angles or alternate interior angles are congruent, or consecutive interior angles are supplementary (sum to 180°).

2. **Equation Setup**: 
   - Using the consecutive interior angle relationship: The angle \( 16x^\circ \) and \( 28x^\circ + 48^\circ \) should sum up to 180° because they are consecutive interior angles.
   
   Thus, we can write:
   \[
   16x + (28x + 48) = 180
   \]
   
3. **Solve for \( x \)**:
   \[
   16x + 28x + 48 = 180
   \]
   \[
   44x + 48 = 180
   \]
   \[
   44x = 180 - 48
   \]
   \[
   44x = 132
   \]
   \[
   x = \frac{132}{44} = 3
   \]

So, the value of \( x \) is 3.

### Converse Used:
The converse of the **Consecutive Interior Angles Theorem** is used: *If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel*.

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Would you like more details or have any other questions? Here are a few related questions for further exploration:

1. How do corresponding angles help determine if lines are parallel?
2. What is the difference between alternate interior and alternate exterior angles?
3. Can the angles formed by two parallel lines and a transversal be congruent instead of supplementary?
4. How does the Converse of the Alternate Interior Angles Theorem apply in such problems?
5. What are other ways to prove two lines are parallel without using angles?

**Tip**: Always remember that for parallel lines, the relationship between angles is key to determining whether lines are truly parallel!

Find the value of x so that line l is parallel to line m. State the converse used.

In this geometry problem, find the value of x that makes two lines parallel. The problem involves parallel lines cut by a transversal and solving for the value of x using angle relationships, specifically the converse of the Consecutive Interior Angles Theorem.

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