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To prove that triangles \( \triangle ABD \) and \( \triangle AEC \) are congruent based on the given information, let's go through the steps.

### Given:
1. \( AB \cong AE \)
2. \( \angle 1 \cong \angle 2 \)

### To Prove:
\( \triangle ABD \cong \triangle AEC \)

### Proof:
We'll use the ASA (Angle-Side-Angle) congruence criterion if possible since we have one angle and a side given.

1. **Identify the shared angle \( \angle A \):**
   - Both \( \triangle ABD \) and \( \triangle AEC \) share \( \angle A \), so \( \angle A \cong \angle A \) by the Reflexive Property.

2. **Given congruent sides \( AB \cong AE \):**
   - This is one of the conditions provided in the problem, indicating that these sides are congruent.

3. **Given congruent angles \( \angle 1 \cong \angle 2 \):**
   - The problem specifies that \( \angle 1 \) (in \( \triangle ABD \)) is congruent to \( \angle 2 \) (in \( \triangle AEC \)).

4. **Conclusion using ASA:**
   - We now have two angles and the included side congruent between \( \triangle ABD \) and \( \triangle AEC \):
     - \( \angle 1 \cong \angle 2 \),
     - \( AB \cong AE \), and
     - \( \angle A \cong \angle A \).
   - By the Angle-Side-Angle (ASA) postulate, \( \triangle ABD \cong \triangle AEC \).

Thus, we have proved that \( \triangle ABD \cong \triangle AEC \).

Would you like more details or have any questions?

### Related Questions:
1. Can we use any other congruence postulate to prove \( \triangle ABD \cong \triangle AEC \)?
2. What would change if \( \angle 3 \) were given as congruent to \( \angle F \)?
3. Is it possible to prove the triangles congruent using the SSS (Side-Side-Side) postulate?
4. How would the proof change if we knew \( BD \cong EC \)?
5. What role does the point \( F \) play in the congruence of the triangles?

### Tip:
Always look for shared angles or sides when trying to prove congruence between two triangles; they often simplify the proof process significantly.

Given: AB ≅ AE, ∠1 ≅ ∠2. Prove: ΔABD ≅ ΔAEC

Learn how to prove the congruence of triangles ΔABD and ΔAEC using the ASA postulate. This geometry problem provides step-by-step reasoning involving congruent angles and sides. Suitable for high school students in Grades 9-10.

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