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Let's analyze and complete the proof from your worksheet about congruence in overlapping triangles. The problem is asking to prove that \( AD \cong BE \), given that \( \triangle AFD \) and \( \triangle BGE \) are equilateral triangles, and some other information such as \( \angle A \cong \angle B \), \( DE \cong FG \), and the other details shown.

Here's a breakdown of the proof statements and reasons:

### Statements:
1. \( \triangle AFD \) and \( \triangle BGE \) are equilateral triangles.
2. \( \angle A \cong \angle B \).
3. \( \angle DAF \cong \angle EBG \).
4. \( DE \cong FG \).
5. \( \triangle AFD \cong \triangle BGE \) by ___ (Need reasoning).
6. \( EF \cong EF \) (Common side).
7. \( DE + EF \cong EF + FG \).
8. \( AD \cong BE \).

### Reasons:
1. Given.
2. Corresponding angles in equilateral triangles.
3. Given (Equilateral triangles have all equal angles).
4. Given.
5. ___ (This is where a method like **AAS** or **SAS** congruence comes into play).
6. Reflexive property of congruence.
7. Segment Addition Postulate.
8. Proven from triangle congruence.

The missing parts can be filled with:
- **Reason for Step 5**: \( \triangle AFD \cong \triangle BGE \) by **AAS (Angle-Angle-Side)**. Two angles and a non-included side are congruent.
- **Reason for Step 11**: Conclude by the definition of congruent triangles, since \( AD \cong BE \) as corresponding parts of congruent triangles.

Would you like a deeper explanation of any specific part of the proof?

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### Related Questions:
1. What does it mean for two triangles to be congruent by AAS?
2. How do you determine corresponding angles in equilateral triangles?
3. What is the significance of the reflexive property in geometry?
4. How does the Segment Addition Postulate work?
5. What other triangle congruence postulates could be applied here?

### Tip:
When dealing with overlapping triangles, always try to separate and redraw them if necessary. This can make it easier to identify corresponding angles and sides.

Prove that AD ≅ BE given that ∆AFD and ∆BGE are equilateral triangles.

Learn how to prove that AD is congruent to BE by applying the AAS congruence theorem and the Segment Addition Postulate in overlapping equilateral triangles. This problem is suitable for high school students studying triangle congruence proofs.

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