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The problem shows that triangles \( \triangle DEF \) and \( \triangle DLK \) are similar, denoted by \( \triangle DEF \sim \triangle DLK \). Since these two triangles are similar, their corresponding sides are proportional.

We are given the following side lengths:
- \( DE = 5 \), \( DF = 2x \), and \( EF = 5 \) for \( \triangle DEF \)
- \( DL = 3 \), \( DK = 6 \), and \( LK = 3 \) for \( \triangle DLK \)

### Corresponding side pairs from the similar triangles:
- \( \frac{DE}{DL} = \frac{5}{3} \)
- \( \frac{DF}{DK} = \frac{2x}{6} \)
- \( \frac{EF}{LK} = \frac{5}{3} \)

Since these triangles are similar, all the ratios of corresponding sides are equal, so:
\[
\frac{DF}{DK} = \frac{DE}{DL}
\]
Substitute the known values:
\[
\frac{2x}{6} = \frac{5}{3}
\]

Now, solve for \( x \).

\[
\frac{2x}{6} = \frac{5}{3}
\]

First, simplify the fraction on the left:
\[
\frac{x}{3} = \frac{5}{3}
\]

Since the denominators are the same, set the numerators equal:
\[
x = 5
\]

Thus, \( x = 5 \).

Would you like further details, or do you have any questions? Here are 5 related questions to consider:

1. How can you check if two triangles are similar using angle criteria?
2. What other properties of similar triangles can be useful in solving geometric problems?
3. How do you find unknown angles in similar triangles using proportional sides?
4. How is the concept of similarity applied in real-world problems, such as map scaling or architecture?
5. Can similar triangles be used to determine unknown heights or distances?

**Tip:** Always double-check the correspondence of sides when working with similar triangles to avoid mismatches in proportions.

Given two similar triangles \( \triangle DEF \sim \triangle DLK \), with side lengths for \( DE = 5 \), \( DF = 2x \), \( EF = 5 \), \( DL = 3 \), \( DK = 6 \), and \( LK = 3 \), find the value of \( x \) using the properties of similar triangles.

Learn how to solve for x using the properties of similar triangles in this geometry problem. Given two triangles \( \triangle DEF \sim \triangle DLK \), with known side lengths, we use proportionality to find the unknown value of x. Suitable for students in Grades 8-10.

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