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It is given that AB=AD=AE, Angle ABC equals Angle ADC, and that D is a point on CE. Prove that triangles ABC and AEC have two sides and an angle in front of the shortest side which are respectively equal, but the triangles aren't congruent.

To prove that triangles \( \triangle ABC \) and \( \triangle AEC \) have two sides and an angle in front of the shortest side which are respectively equal but the triangles aren't congruent, let's proceed with the following steps:

### Given Information:
1. \( AB = AD = AE \)
2. \( \angle ABC = \angle ADC \)
3. \( D \) is a point on \( CE \).

### Proof:
1. **Identify the two triangles**:
   - We are working with \( \triangle ABC \) and \( \triangle AEC \).

2. **Check for equal sides**:
   - \( AB = AE \) (given)
   - \( AC \) is common to both triangles \( \triangle ABC \) and \( \triangle AEC \).

3. **Check for equal angles**:
   - \( \angle ABC = \angle ADC \) (given)
   - Notice that \( \angle ABC \) is part of \( \triangle ABC \) and \( \angle ADC \) is part of \( \triangle AEC \).

4. **Side-Angle-Side (SAS) similarity**:
   - For triangles to have two sides and the included angle equal, the triangles might be similar by SAS (Side-Angle-Side) similarity.
   - In \( \triangle ABC \) and \( \triangle AEC \):
     - \( AB = AE \) (equal sides)
     - \( AC \) is common (equal sides)
     - \( \angle BAC \) (included angle) is common in both triangles \( \triangle ABC \) and \( \triangle AEC \).

   Thus, \( \triangle ABC \) and \( \triangle AEC \) have two sides equal and the angle between them equal.

5. **Non-congruence**:
   - The triangles are not congruent because \( BC \) is not equal to \( EC \). In \( \triangle ABC \), \( BC \) is a side, and in \( \triangle AEC \), \( EC \) is a side.
   - Since \( BC \) and \( EC \) are different, the two triangles cannot be congruent.

### Conclusion:
- Triangles \( \triangle ABC \) and \( \triangle AEC \) have two sides and the included angle respectively equal. However, they are not congruent because their third sides \( BC \) and \( EC \) are not equal.

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. What is the significance of the angle equality \( \angle ABC = \angle ADC \) in proving similarity or congruence?
2. Can we prove that triangles \( \triangle ABC \) and \( \triangle AEC \) are similar, and if so, under what criteria?
3. How would the proof change if \( AB \) were not equal to \( AE \)?
4. What role does the point \( D \) on \( CE \) play in this geometric configuration?
5. What other geometric properties can be deduced from this configuration?

**Tip:** When dealing with triangles, remember that congruence implies similarity, but similarity does not necessarily imply congruence.

Explore the proof involving triangles ABC and AEC with equal sides AB = AD = AE and angles ABC = ADC. Understand how these properties indicate similarity without congruence, highlighting geometric principles.

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