To solve this problem using the concept of similar triangles:

1. Given:

   • $EF \parallel BD$ and $DE \parallel BC$.
   • Triangles are divided into similar sections due to parallel lines.

2. Key Observations:

   • Triangle $\triangle AEF$ is similar to $\triangle ABD$ (by AA similarity because $EF \parallel BD$ and corresponding angles are equal).
   • $\triangle ADE$ is similar to $\triangle ABC$ (for the same reasoning).

3. Similar Triangle Ratios:

   • For $\triangle AEF \sim \triangle ABD$: $\frac{AE}{AB} = \frac{AF}{AD} = \frac{EF}{BD}$ Let us use $AE = 8$, $EF = x$, $BD = 6$, and find the other side ratio.

   • For $\triangle ADE \sim \triangle ABC$: $\frac{AE}{AB} = \frac{DE}{BC}$ Here, $DE = x$, and $BC = 6$.

4. Setting Up Equations:
   From the proportional relationships: $\frac{AE}{AB} = \frac{8}{8 + 6} = \frac{8}{14} = \frac{4}{7}.$

   Substitute into the ratios:

   • $\frac{EF}{BD} = \frac{x}{6} = \frac{4}{7}$, solving for $x$: $x = \frac{4}{7} \times 6 = \frac{24}{7} \approx 3.43.$

Final Answer:

$x = \frac{24}{7} \text{ or approximately } 3.43.$

Let me know if you need further details or clarification!

Related Questions:

1. How do we identify similar triangles in geometric problems?
2. Can you prove that $\triangle ADE \sim \triangle ABC$ without parallel lines?
3. Why is proportionality of sides important in similar triangles?
4. How would the solution change if $EF \neq BD$?
5. What other methods can be used to find $x$ in this problem?

Tip:

When solving similarity problems, always double-check for parallel lines and equal angles—they often hint at similar triangles.