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To help you with this geometry proof, let’s analyze the given information and the steps in the proof.

### Given:
1. \( BC \cong AD \): Segment \( BC \) is congruent to segment \( AD \).
2. \( BC \parallel AD \): Segment \( BC \) is parallel to segment \( AD \).

### To Prove:
\[ \triangle BEC \cong \triangle DEA \]

### Proof Steps Analysis:
| Step | Statement                | Reason                                     |
|------|---------------------------|--------------------------------------------|
| 1    | \( BC \cong AD \), \( BC \parallel AD \) | Given                              |
| 2    | \( \angle BEC \cong \angle DEA \)        | Vertical angles are congruent       |
| 3    | \( AC \cong AC \)                        | Reflexive Property                  |
| 4    | \( BD \cong BD \)                        | Reflexive Property                  |
| 5    | \( \angle AEB \cong \angle DEC \)        | Vertical angles are congruent       |

### Completion and Justification:
Given that we need to prove \( \triangle BEC \cong \triangle DEA \), let’s apply the **ASA (Angle-Side-Angle) Congruence Postulate** as follows:

- **Angle 1**: \( \angle BEC \cong \angle DEA \) (Vertical angles are congruent).
- **Side**: \( BC \cong AD \) (Given).
- **Angle 2**: \( \angle AEB \cong \angle DEC \) (Vertical angles are congruent).

Since we have two angles and the included side congruent between \( \triangle BEC \) and \( \triangle DEA \), we conclude by the ASA postulate that:
\[ \triangle BEC \cong \triangle DEA \]

Would you like more details on any specific step or concept?

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### Additional Related Questions:
1. What is the significance of the ASA Congruence Postulate in triangle congruence?
2. How does the Reflexive Property help in proving triangles congruent?
3. What are other triangle congruence postulates or theorems (e.g., SSS, SAS) and how are they used?
4. Can we use the concept of parallel lines to prove congruence of other triangles in different ways?
5. Why are vertical angles always congruent?

### Tip:
In geometry proofs, clearly stating each reason for congruence or equality (e.g., Reflexive Property, ASA Postulate) can make your proof more understandable and help ensure correctness.

Given: BC ≅ AD and BC ∥ AD. Prove: △BEC ≅ △DEA.

Explore a detailed proof of triangle congruence for △BEC and △DEA using the ASA Congruence Postulate. This geometry problem covers concepts like vertical angles, parallel lines, and congruent segments, ideal for high school students in Grades 9-10.

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