This image contains a geometry proof task. The goal is to prove that triangles $\triangle ABE \cong \triangle CDE$, under the given condition that $BD$ and $AC$ bisect each other. Let's complete the proof step-by-step.

Given:

• $BD$ and $AC$ bisect each other (Given).
• $\angle ABE \cong \angle CDE$ (Vertical angles are congruent).

Proof:

We aim to prove $\triangle ABE \cong \triangle CDE$.

Steps for Proof:

1. Statement: $BD$ and $AC$ bisect each other.
   Reason: Given.

2. Statement: $\angle ABE \cong \angle CDE$.
   Reason: Vertical angles are congruent.

3. Statement: $AE \cong CE$ and $BE \cong DE$.
   Reason: Since $BD$ and $AC$ bisect each other, the segments are divided into equal halves.

4. Statement: $\triangle ABE \cong \triangle CDE$.
   Reason: By the SAS (Side-Angle-Side) congruence postulate.

   • The sides $AE$ and $BE$ of $\triangle ABE$ are congruent to $CE$ and $DE$, respectively.
   • The angle $\angle ABE \cong \angle CDE$ is included between these two sides.

Thus, $\triangle ABE \cong \triangle CDE$ is proved.

Would you like a more detailed explanation of any specific step or concept? Here are some follow-up questions to expand your understanding:

1. What does it mean for two triangles to be congruent?
2. How do vertical angles arise in this proof, and why are they congruent?
3. What other congruence postulates or theorems could apply here (e.g., ASA, SSS)?
4. How can the concept of bisection in a line segment be formally defined?
5. Could this proof change if we introduced coordinate geometry or a numerical example?

Tip:

When proving triangle congruence, always identify the congruence postulate (SAS, SSS, ASA, or AAS) to ensure your reasoning is structured and complete!