https://file.aiquickdraw.com/mathbot/file/1729087449833o034hucl.jpg

To solve for \(x\) in the given problem, we need to apply the property that similar triangles have proportional sides. The triangles given are \(\triangle DEF \sim \triangle LMN\).

Let's set up a proportion based on corresponding sides of the triangles:
- \(DE\) corresponds to \(LM\)
- \(EF\) corresponds to \(MN\)
- \(DF\) corresponds to \(LN\)

From the image:
- \(DE = 15\)
- \(EF = 11 - x\)
- \(LM = 5\)
- \(MN = 4\)
- \(LN = 3\)

We can use the proportionality of sides between the similar triangles:

\[
\frac{DE}{LM} = \frac{EF}{MN}
\]

Substitute the values:

\[
\frac{15}{5} = \frac{11 - x}{4}
\]

Simplify the left-hand side:

\[
3 = \frac{11 - x}{4}
\]

Now, solve for \(x\):

1. Multiply both sides by 4 to eliminate the denominator:

\[
12 = 11 - x
\]

2. Subtract 11 from both sides:

\[
1 = -x
\]

3. Multiply by -1:

\[
x = -1
\]

Thus, the value of \(x\) is \(-1\).

Would you like further explanation or have any questions?

### Related Questions:
1. How do you prove triangles are similar using the AA (Angle-Angle) postulate?
2. What other properties can you use when triangles are similar?
3. Can you solve for other unknowns if one of the sides in triangle \(\triangle LMN\) was missing?
4. What happens if one of the angles is missing instead of the sides?
5. How does the Pythagorean Theorem apply to similar triangles with right angles?

### Tip:
In similar triangles, always use corresponding sides to set up proportions accurately for solving unknowns.

Solve for x in the problem where ΔDEF is similar to ΔLMN.

This problem demonstrates how to solve for x using the property of proportionality in similar triangles. Learn step-by-step how corresponding sides of ΔDEF and ΔLMN are used to set up and solve equations. Suitable for students in Grades 9-11.

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