Let's continue the proof based on the information provided in the image.

Here's what we have so far:

1. Given: $BD$ and $AC$ bisect each other.
2. Statement: $AE \cong EC$ and $BE \cong ED$
   • Reason: A segment bisector divides a segment into two congruent segments.
3. Statement: $\angle AEB \cong \angle CED$
   • Reason: Vertical angles are congruent.

We want to prove that $AB \parallel CD$.

Next Steps to Complete the Proof

4. Statement: $\triangle AEB \cong \triangle CED$

   • Reason: Side-Angle-Side (SAS) Congruence Postulate, because $AE \cong EC$, $\angle AEB \cong \angle CED$, and $BE \cong ED$.

5. Statement: $\angle BAE \cong \angle DCE$

   • Reason: Corresponding parts of congruent triangles are congruent (CPCTC).

6. Statement: $AB \parallel CD$

   • Reason: If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

This completes the proof that $AB \parallel CD$.

Would you like a more detailed explanation of each step?

Here are some related questions to deepen understanding:

1. Why do vertical angles always turn out to be congruent?
2. How does the SAS Postulate help in proving triangles congruent?
3. What role does CPCTC play in geometric proofs?
4. Can there be other ways to prove $AB \parallel CD$ without using congruent triangles?
5. What other properties of bisected segments can help in proofs of parallel lines?

Tip: Remember that in geometry, segment bisectors and angle relationships are powerful tools to establish congruence and parallelism.