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This image shows a diagram with three parallel lines intersected by two transversals. The line segments on these transversals have lengths labeled, and the relationship between them can be solved using the basic concept of proportionality of corresponding segments formed by parallel lines and transversals.

The given information is:
- A segment of length 9, another of length 4, a third labeled \( 13.5 \), and the last segment labeled \( x + 4 \).

These lengths belong to two transversals cut by parallel lines, so the ratios of the corresponding segments will be equal. This allows us to set up the following proportion:

\[
\frac{9}{4} = \frac{13.5}{x + 4}
\]

Now, solving for \( x \):

1. Cross multiply:
\[
9(x + 4) = 4 \times 13.5
\]

2. Simplify the right-hand side:
\[
9(x + 4) = 54
\]

3. Distribute the 9 on the left:
\[
9x + 36 = 54
\]

4. Subtract 36 from both sides:
\[
9x = 18
\]

5. Divide both sides by 9:
\[
x = 2
\]

Thus, \( x = 2 \).

Would you like more details or have any other questions?

Here are 5 related questions:
1. How can you check if two line segments are proportional?
2. What are the conditions for two lines to be considered parallel?
3. What is the significance of transversals in geometry?
4. How do you solve proportions with variables in both the numerator and denominator?
5. How can we apply this concept of ratios in other geometric shapes, like triangles?

**Tip:** When dealing with parallel lines and transversals, remember that corresponding segments are proportional, which is key in many geometric proofs.

In the given diagram of parallel lines intersected by two transversals, solve for the unknown value 'x' given that the lengths of corresponding segments are proportional. One transversal has segments labeled 9 and 4, while the other has segments labeled 13.5 and x+4.

This geometry problem involves solving for the unknown 'x' by applying the concept of proportionality between corresponding segments of transversals intersecting parallel lines. The problem uses a setup with given segment lengths of 9, 4, 13.5, and x + 4, leading to a step-by-step algebraic solution.

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