To solve for $x$, let’s analyze the triangle and the given expressions:

Information from the diagram:

1. $\triangle ABC$ is an isosceles triangle (indicated by the congruent sides $AC \cong BC$).
2. Segment $DE$ is parallel to $AB$, dividing $\triangle ABC$ proportionally.
3. The proportionality rule applies here, as the smaller triangle $\triangle CDE$ is similar to the larger triangle $\triangle ABC$.

Given lengths:

• $AB = 2x + 4$ (base of the larger triangle),
• $DE = 7x - 82$ (base of the smaller triangle),
• $AC \cong BC$ ensures the similarity applies.

Using the property of similar triangles:

The ratio of the corresponding sides of similar triangles is equal. Specifically: $\frac{DE}{AB} = \frac{CD}{CA}.$

Since the diagram shows that $CD \cong CE$ and $DE \parallel AB$, the similarity simplifies the problem to: $\frac{DE}{AB} = \frac{CD}{CA}.$

Substituting the given values:

Using proportional similarity, we write: $\frac{7x - 82}{2x + 4} = \frac{CD}{CA}.$ We would need clarification for further values (e.g., a specific ratio or segment lengths) to solve completely.

Let me know if there’s further information about this triangle or if you’d like to clarify the proportionality ratio!