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The problem involves two triangles: \( \triangle LMN \) and \( \triangle DEF \), which appear to be similar by the Angle-Angle (AA) criterion since both triangles have an angle of \( 86^\circ \). The task is to prove the similarity and find the value of \( x \).

### Step-by-step Solution:

1. **Similarity Criterion (SSS):**
   - The triangles are given as similar by Side-Side-Side (SSS), as indicated by the reason in the image. This suggests that corresponding sides are proportional.

2. **Set up the proportion between corresponding sides:**
   - In \( \triangle LMN \) and \( \triangle DEF \), the sides should be proportional.
   - The corresponding sides are:
     - \( \frac{LM}{DF} = \frac{9}{4} \),
     - \( \frac{MN}{EF} = \frac{4x - 10}{x - 1} \),
     - \( \frac{LN}{DE} = \frac{12}{3} = 4 \).

3. **Solving for \( x \):**
   - From the proportionality, we can use:
     \[
     \frac{MN}{EF} = \frac{LN}{DE}
     \]
     which gives the equation:
     \[
     \frac{4x - 10}{x - 1} = 4
     \]
   - Solve the equation:
     \[
     4x - 10 = 4(x - 1)
     \]
     \[
     4x - 10 = 4x - 4
     \]
     \[
     -10 = -4
     \]
   - There seems to be no solution based on this proportion, so let's recheck the proportion logic.

4. **Correcting proportionality:**
   - Alternatively, using the correct sides might resolve the proportion:
     \[
     \frac{MN}{EF} = \frac{LM}{DF}
     \]
     leading to:
     \[
     \frac{4x - 10}{x - 1} = \frac{9}{4}
     \]
   - Cross-multiply:
     \[
     4(4x - 10) = 9(x - 1)
     \]
     \[
     16x - 40 = 9x - 9
     \]
     \[
     16x - 9x = 40 - 9
     \]
     \[
     7x = 31
     \]
     \[
     x = \frac{31}{7} \approx 4.43
     \]

### Final Answer:
- The triangles are similar by SSS.
- The value of \( x \) is \( \frac{31}{7} \) or approximately 4.43.

Do you want more details or have any questions about the steps?

Here are 5 related questions for further practice:
1. How can we determine if two triangles are similar by AA criterion?
2. If two triangles are similar, how do we calculate missing side lengths using proportionality?
3. Can SSS similarity be used when angles are unknown, and how?
4. What are the different ways to prove that two triangles are similar?
5. How can you apply triangle similarity in real-life applications like map scaling?

**Tip:** Always double-check proportionality setups by ensuring you correctly match corresponding sides!

Prove the similarity of triangles LMN and DEF, and solve for x using the given side lengths and angles.

Learn how to prove the similarity of triangles LMN and DEF using the SSS similarity criterion and solve for x with proportionality. This problem involves core geometric concepts and algebraic manipulation, suitable for students in Grades 8-10.

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