Given that $AE \cong EC$ and $BE \cong ED$, we want to prove that $AB \cong DC$. We need to determine the appropriate statement for the next step in the proof.

1. Problem Description

Given that $AE \cong EC$ and $BE \cong ED$, we want to prove that $AB \cong DC$. We need to determine the appropriate statement for the next step in the proof.

2. Solution Steps

We are given $AE \cong EC$ and $BE \cong ED$. We can see that $\angle AEB$ and $\angle CED$ are vertical angles.

Vertical angles are congruent.

So, $\angle AEB \cong \angle CED$.

We now have two sides and the included angle congruent in $\triangle AEB$ and $\triangle CED$. Therefore, $\triangle AEB \cong \triangle CED$ by the SAS congruence postulate.

If $\triangle AEB \cong \triangle CED$, then corresponding parts of congruent triangles are congruent (CPCTC).

So, $AB \cong DC$.

The statement that appears to be the most relevant after the given information is that $\angle AEB \cong \angle CED$.

The reason should be the vertical angles theorem.

However, the options listed in the image do not seem to include stating angle congruence. They jump directly to stating that triangles are congruent.

In that case, we can conclude $\triangle AEB \cong \triangle CED$. The reason for this would be SAS congruence.

Then, we can conclude $AB \cong DC$ by CPCTC.

Looking at the given choices, $\triangle XYZ \cong \triangle TUV$ is the best match. The correct statement would be $\triangle AEB \cong \triangle CED$.

3. Final Answer

$\triangle AEB \cong \triangle CED$