To solve this problem, we need to utilize the properties of parallel lines and transversals, specifically the principle that corresponding segments between parallel lines are proportional. Let’s break it down.

Step-by-step Solution:

1. The three parallel lines are labeled $l$, $m$, and $n$, and the transversal lines divide them into segments.
2. From the given figure, the segments on the transversal lines are labeled:
   • On $m$, the segments are $16$ and $x$,
   • On $n$, the segments are $23$ and $38$.

Using the proportionality rule for parallel lines and transversals: $\frac{\text{segment on line } m}{\text{segment on line } n} = \frac{\text{segment on line } m}{\text{segment on line } n}.$

Substitute the values: $\frac{16}{23} = \frac{x}{38}.$

3. Cross-multiply to solve for $x$: $16 \cdot 38 = 23 \cdot x.$

$608 = 23x.$

4. Solve for $x$: $x = \frac{608}{23}.$

$x \approx 26.4.$

Final Answer:

The value of $x$ is approximately 26.4.

Would you like additional details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the proportionality rule work for segments between parallel lines and transversals?
2. What happens if there are more than two transversals?
3. How would the solution change if one segment length was unknown in a different part of the figure?
4. Can this proportionality rule apply if the lines are not parallel? Why or why not?
5. How can this method be extended to solve problems in real-world applications like map scaling?

Tip: Always cross-check calculations when working with proportions to avoid arithmetic errors.