https://file.aiquickdraw.com/mathbot/file/1729066093487dr2qofhz.jpg

In this problem, you are asked to identify the similar triangles and find the value of \( x \). The triangles given are \( \triangle VWX \) and \( \triangle YZX \), which are stated to be similar (denoted by the similarity symbol \( \sim \)).

### Step 1: Use the Property of Similar Triangles
Since the triangles \( \triangle VWX \) and \( \triangle YZX \) are similar, corresponding sides are proportional. Therefore, we can set up a proportion using the corresponding sides of the triangles.

- Corresponding sides are:
  \[
  \frac{VW}{YZ} = \frac{WX}{ZX} = \frac{VX}{YX}
  \]

From the diagram, the lengths of the sides are:
- \( VW = 2 \)
- \( YZ = 8 \)
- \( WX = 3 \)
- \( ZX = 2x + 4 \)

### Step 2: Set up the Proportion
We can use the sides \( VW \) and \( YZ \), and \( WX \) and \( ZX \) for the proportion:
\[
\frac{VW}{YZ} = \frac{WX}{ZX}
\]

Substituting the known values:
\[
\frac{2}{8} = \frac{3}{2x + 4}
\]

### Step 3: Solve for \( x \)
Now, cross-multiply to solve for \( x \):
\[
2 \cdot (2x + 4) = 8 \cdot 3
\]
\[
2(2x + 4) = 24
\]
\[
4x + 8 = 24
\]
\[
4x = 16
\]
\[
x = 4
\]

Thus, the value of \( x \) is \( \boxed{4} \).

Would you like more details or explanations about any steps? Here are some related questions for deeper exploration:

1. How can you determine that two triangles are similar using the AA (Angle-Angle) criterion?
2. What are the different ways to prove triangle similarity (AA, SSS, SAS)?
3. How do you solve proportions involving variables?
4. Why do corresponding sides of similar triangles have proportional lengths?
5. How can triangle similarity help in real-world applications, such as indirect measurement?

**Tip:** When working with similar triangles, always make sure to correctly match corresponding sides and angles before setting up proportions.

Identify the similar triangles and find the value of x in the given triangle problem with triangles VWX and YZX.

Learn how to solve for x in the geometry problem involving similar triangles VWX and YZX. This problem demonstrates the use of the AA similarity theorem and setting up proportions to solve for unknown sides. Suitable for students in Grades 8-10.

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