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The problem shows two overlapping triangles, \( \triangle ABE \) and \( \triangle CBD \), and the question asks which congruence relationship proves that these triangles are congruent.

From the diagram and the given information:
1. \( \angle AEB \cong \angle CDB \) (the two angles marked congruent at the base of each triangle).
2. \( \overline{EB} \cong \overline{DB} \) (the sides between these angles are marked as congruent).
3. \( \overline{AB} \) and \( \overline{CB} \) are shared sides, so they are congruent by the reflexive property.

The most appropriate congruence criterion in this case is **ASA (Angle-Side-Angle)**:
- One pair of angles is congruent (\( \angle AEB \cong \angle CDB \)),
- The side between those angles (\( \overline{EB} \cong \overline{DB} \)),
- The included side \( \overline{AB} \cong \overline{CB} \) by reflexivity.

Thus, the answer is **ASA**.

Would you like a detailed explanation on triangle congruence, or do you have any further questions? Here are some related questions:

1. What other conditions for triangle congruence are commonly used?
2. Why is the reflexive property important in congruence problems?
3. How does the ASA criterion differ from SAS or SSS?
4. Can you give an example of using the HL (Hypotenuse-Leg) theorem?
5. How do we prove congruence in non-right triangles?

**Tip:** When solving geometry problems, always look for shared sides or angles to help establish congruence or similarity between triangles.

Which of the following relationships proves why ΔABE and ΔCBD are congruent?

Explore how to prove the congruence of triangles ΔABE and ΔCBD using the ASA (Angle-Side-Angle) congruence theorem. This problem involves overlapping triangles and the use of the reflexive property, suitable for students in Grades 8-10.

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