The question requires proving that triangles $\triangle BEC$ and $\triangle DEA$ are congruent, given two key pieces of information:

1. $BC \cong AD$
2. $BC \parallel AD$

The proof steps provided include vertical angles and reflexive properties, but it appears the final step in the congruence proof has been left incomplete, as the attempt to use the reflexive property on sides $BA$ and $CD$ seems to be incorrect.

Let's go over the necessary steps to prove $\triangle BEC \cong \triangle DEA$:

Steps to Complete the Proof

1. Given Information:

   • $BC \cong AD$ (Given)
   • $BC \parallel AD$ (Given)

2. Vertical Angles:

   • $\angle BEC \cong \angle DEA$ because they are vertical angles, which are always congruent.

3. Reflexive Property:

   • To satisfy the criteria of the Side-Angle-Side (SAS) Congruence Postulate, identify a shared side or additional information. Here:
     • Since triangles $\triangle BEC$ and $\triangle DEA$ share side $BE$ and $DE$, we have $BE \cong DE$ by the Reflexive Property.

4. Conclusion Using SAS:

   • By the Side-Angle-Side (SAS) postulate, $\triangle BEC \cong \triangle DEA$, as we have:
     • $BC \cong AD$ (side)
     • $\angle BEC \cong \angle DEA$ (angle)
     • $BE \cong DE$ (side by Reflexive Property)

Correction of the Final Step

The final missing step to complete the proof should involve applying the SAS Congruence Postulate, not using the reflexive property on unrelated sides like $BA$ and $CD$.

Would you like further details on any part of this explanation?

Additional Questions to Deepen Understanding

1. How does the parallelism between $BC$ and $AD$ contribute to the congruence of triangles?
2. What other congruence postulates might be applied if different information were given?
3. Why are vertical angles always congruent?
4. How does the reflexive property help in congruence proofs?
5. Could this proof be done using any other congruence criteria (e.g., ASA, SSS)?

Tip

When proving congruence, always look for shared sides or angles as they often help complete the proof by SAS or ASA.