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The question asks to prove that triangle \( \triangle ADE \) in figure 2 is an isosceles triangle. Let's break this down.

We are given:
- In both figures, \( \triangle ABC \) is an isosceles triangle, where \( AB = AC \), and angle \( A \) is greater than \( 90^\circ \).
- In figure 2, points \( D \) and \( E \) are selected on side \( BC \) such that \( BD = CE \).

### Step-by-step proof:

1. **Given Conditions**:
   - \( AB = AC \) (from the problem).
   - \( BD = CE \) (from figure 2).

2. **Segment Lengths**:
   - \( D \) and \( E \) are located such that the two segments on the side \( BC \) are equal, i.e., \( BD = CE \). This divides side \( BC \) into equal segments.

3. **Triangles \( \triangle ABD \) and \( \triangle ACE \)**:
   - Since \( AB = AC \) and \( BD = CE \), by the **Side-Side-Side (SSS) congruence**, the triangles \( \triangle ABD \) and \( \triangle ACE \) are congruent.

4. **Angle Equality**:
   - Because \( \triangle ABD \cong \triangle ACE \), the angles \( \angle ADB \) and \( \angle AEC \) are equal.

5. **Isosceles Triangle \( \triangle ADE \)**:
   - Since the angles \( \angle ADB = \angle AEC \), and the base segments \( BD \) and \( CE \) are equal, this symmetry shows that triangle \( \triangle ADE \) must be isosceles, with \( AD = AE \) by the congruence of the smaller triangles.

Therefore, \( \triangle ADE \) is an isosceles triangle as required.

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Would you like more details on any specific part of the proof or explanation?

Here are 5 related questions you could explore:
1. How does the angle-sum property of a triangle help in proving properties of isosceles triangles?
2. Can congruent triangles have different angles, or are the angles always the same?
3. How do equal segments on a side help in proving other geometric properties of a triangle?
4. What role does the inequality of \( \angle A \) (being greater than \( 90^\circ \)) play in this problem?
5. How would the proof change if \( AB \neq AC \)?

**Tip**: Whenever dealing with isosceles triangles, checking symmetry and congruence is key to proving equal sides and angles.

Prove that triangle ADE in figure 2 is an isosceles triangle, where ABC is an isosceles triangle with AB = AC, angle A > 90°, and points D and E on BC such that BD = CE.

This geometry problem involves proving that triangle ADE is isosceles using the Side-Side-Side (SSS) congruence rule and symmetry in triangle ABC. Given that AB = AC and BD = CE, the problem demonstrates how to use triangle congruence to prove properties of isosceles triangles. Suitable for students in Grades 9-10.

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