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To prove that triangles \( \triangle BEC \cong \triangle DEA \), let's go through the steps based on the given information and the properties of congruent triangles.

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

### Proof Steps:

1. **Statement**: \( BC \cong AD \)  
   **Reason**: Given.

2. **Statement**: \( BC \parallel AD \)  
   **Reason**: Given.

3. **Statement**: \( \angle BEC \cong \angle DEA \)  
   **Reason**: Alternate interior angles theorem (since \( BC \parallel AD \) and \( BE \) and \( ED \) are transversals).

4. **Statement**: \( BE \cong DE \)  
   **Reason**: Shared side in both \( \triangle BEC \) and \( \triangle DEA \).

5. **Conclusion**: \( \triangle BEC \cong \triangle DEA \)  
   **Reason**: **SAS (Side-Angle-Side) Congruence Theorem** (since we have two sides and the included angle congruent in both triangles).

### Explanation:
By establishing that \( BC \parallel AD \) and using the alternate interior angles formed by the transversal \( BE \) (or \( ED \)), we show that \( \angle BEC \cong \angle DEA \). Additionally, since \( BE = DE \) (shared side), we can conclude the triangles are congruent by the SAS Congruence Theorem.

Would you like further details or explanations on any specific step?

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### Related Questions:
1. How does the alternate interior angles theorem help in proving congruence?
2. What other properties can be used to prove triangle congruence besides SAS?
3. How does sharing a side help in proving triangle congruence?
4. Can this proof be done using the ASA (Angle-Side-Angle) theorem instead?
5. How does parallelism contribute to proving angles are congruent?

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### Tip:
When working with parallel lines in geometry proofs, remember that alternate interior angles are always congruent, which is helpful for establishing angle congruence in triangles.

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

This geometry proof demonstrates how to use the SAS Congruence Theorem and Alternate Interior Angles Theorem to prove that △BEC ≅ △DEA. Given that BC ≅ AD and BC || AD, follow the step-by-step reasoning to establish triangle congruence for high school geometry.

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