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To find the value of \( x \) in the given diagram, where lines \( l \), \( m \), and \( n \) are parallel and cut by two transversal lines, we can use the property of similar triangles. 

### Step-by-Step Explanation:
1. **Identify Corresponding Angles and Ratios:**
   - Since the lines \( l \), \( m \), and \( n \) are parallel, the triangles formed by the transversals are similar.
   - Therefore, the corresponding sides of the similar triangles are proportional.

2. **Set Up the Proportion:**
   The corresponding sides in the two triangles are proportional, so we can set up the following proportion using the given values:
   \[
   \frac{x}{19} = \frac{46}{56}
   \]
   
3. **Solve for \( x \):**
   \[
   x = 19 \times \frac{46}{56}
   \]
   Simplifying:
   \[
   x = 19 \times 0.8214 = 15.606 \quad (\text{rounded to the nearest tenth})
   \]
   So, \( x \approx 15.6 \).

### Final Answer:
The value of \( x \) is approximately **15.6**.

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Would you like more details or clarifications on the steps? Here are some related questions:

1. How can we confirm that the triangles are similar?
2. Why do the parallel lines allow us to set up a proportion between the triangles?
3. How can we verify the accuracy of the result?
4. What happens if the transversals were not parallel?
5. Can we use angle relationships to approach this problem differently?

**Tip:** When working with similar triangles, always check for corresponding angles and sides to set up accurate proportions.

Given lines l, m, and n are parallel and cut by two transversal lines, find the value of x. Round your answer to the nearest tenth if necessary.

Learn how to solve for x in a geometry problem involving parallel lines and two transversals. This problem covers concepts like similar triangles and proportionality. Suitable for students in Grades 8-10.

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