In this image, we see three parallel lines, $r$, $s$, and $p$, intersected by two transversals. The problem asks for the value of $x$, and we are given the lengths of the segments created by the transversals and the parallel lines:

• On line $r$, the segments are $x$ and $21$.
• On line $s$, the segments are $12$ and $36$.

Since the lines are parallel, we can use the property of similar triangles formed by the transversals intersecting the parallel lines. This means the ratio of corresponding segments on the two lines must be the same.

We set up the following proportion:

$\frac{x}{12} = \frac{21}{36}$

Now, let's solve for $x$.The value of $x$ is 7.

Would you like a detailed explanation of the steps, or do you have any further questions? Here are some related questions for further practice:

1. What property of parallel lines and transversals allows us to set up this proportion?
2. How can you verify that two triangles are similar?
3. What other geometric problems can be solved using proportions from similar triangles?
4. How do parallel lines affect angle relationships in geometry?
5. What is the difference between solving proportions and solving equations?

Tip: When working with similar triangles, always ensure the corresponding sides are correctly matched when setting up proportions.