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The problem involves identifying the correct relationship to prove that triangles \( \triangle ABE \) and \( \triangle CBD \) are congruent. Let’s analyze the given information:

1. **Angle congruences**:  
   \( \angle AEB \cong \angle CDB \).

2. **Side congruences**:  
   \( EB \cong DB \).

3. **Common side**:  
   Both triangles share the side \( \overline{AB} \).

### Solution:
Since:
- **Two angles** are congruent (as stated).
- **The included side** (EB = DB) is congruent between the two triangles.

This corresponds to the **ASA (Angle-Side-Angle)** congruence theorem, which states that two triangles are congruent if two angles and the included side are congruent.

### Correct answer:
**ASA**

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Do you want more details or have any questions?  
Here are some related questions for further exploration:  

1. What other scenarios would use the SAS theorem instead?  
2. How do you apply the SSS theorem to prove congruence?  
3. What are the properties of overlapping triangles in geometry?  
4. How does the HL theorem work for right triangles?  
5. Can you explain the difference between congruence and similarity?

**Tip:** Remember to always identify the included side between two angles when deciding if ASA applies!

Which of the following relationships proves why triangles ABE and CBD are congruent? Given that angle AEB is congruent to angle CDB, and side EB is congruent to side DB.

Learn how to prove that triangles ABE and CBD are congruent using the ASA (Angle-Side-Angle) theorem. This problem demonstrates the application of triangle congruence in overlapping triangles. Suitable for students in Grades 8-10.

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