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Let's solve the given problem step by step. The problem provides the following information:
1. \( DE \parallel AB \)
2. \( CF \parallel AB \)
3. \( BG \parallel AB \)

The goal is to find the length of \( x \).

The diagrams show a set of similar triangles. We can use the properties of similar triangles to solve for \( x \).

Since \( DE \parallel AB \), \( \triangle CDE \) is similar to \( \triangle CBA \).

Since \( CF \parallel AB \), \( \triangle CFB \) is similar to \( \triangle CBA \).

And since \( BG \parallel AB \), \( \triangle BGA \) is similar to \( \triangle CBA \).

Given:
\( DC = 7 \)
\( CB = 3 \)
\( BA = x \)

First, let's write the similarity ratios:
For \( \triangle CDE \sim \triangle CBA \):
\[ \frac{DC}{CB} = \frac{DE}{BA} \]

For \( \triangle CFB \sim \triangle CBA \):
\[ \frac{CF}{BA} = \frac{CB}{BA} \]

For \( \triangle BGA \sim \triangle CBA \):
\[ \frac{BG}{BA} = \frac{BA}{BA} \]

To find \( x \):
\[ \frac{7}{3} = \frac{7}{x} \]

Since \( DE \parallel AB \), all segments are proportional. So, the ratio will be:

\[ \frac{7}{3} = \frac{DE}{x} \]

The problem becomes:

\[ 7 \cdot x = 3 \cdot 7 \]
\[ 7x = 21 \]
\[ x = 3 \]

However, this doesn't align with the problem constraints. To address this, we must correct the interpretation, but let's consider if the proper interpretation involves additional triangles:

### Correct interpretation might involve:
Considering intermediate steps and correct proportions:
\[ \frac{DE}{CF} = \frac{CF}{BG} \]

This forms a valid proportional statement:
\[ \frac{7}{3 + x} = \frac{3}{x} \]
By cross-multiplying:
\[ 7x = 3(3 + x) \]
\[ 7x = 9 + 3x \]
\[ 7x - 3x = 9 \]
\[ 4x = 9 \]
\[ x = \frac{9}{4} \approx 2.25 \]

Hence, based on updated triangles and correctly solving proportion:
\[ x = \frac{9}{4} \]

**Tip**: Always confirm proportional relationships to avoid basic misinterpretations.

### Further Questions:
1. What are properties of similar triangles?
2. Explain AA (Angle-Angle) similarity criterion?
3. Discuss properties of parallel lines?
4. What is cross-multiplication used for?
5. Explain ratios and proportions in triangles?
6. Define parallel line segment properties?
7. What is the method for solving proportion equations?
8. Discuss key steps for validating triangle similarity?

Is any specific solution or steps confusing? Would you like more details?

Explore how to solve proportions in similar triangles using the AA (Angle-Angle) similarity criterion. This problem involves finding the length of a segment in similar triangles where parallel lines create proportional relationships. Suitable for high school students.

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