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To prove that \(\angle EFD \cong \angle GDF\), let's break down the proof step by step, using the given information.

### Given:
1. \( \overline{DE} \cong \overline{FG} \) (The two sides are congruent)
2. \( \angle EDF \cong \angle GFD \) (The two angles are congruent)

### Goal:
Prove that \(\angle EFD \cong \angle GDF\).

We are dealing with two triangles: \(\triangle EFD\) and \(\triangle GDF\). To prove that the two angles \(\angle EFD\) and \(\angle GDF\) are congruent, we'll rely on the given information and apply geometric theorems.

### Plan for the Proof:
We can prove this using the **ASA (Angle-Side-Angle) Congruence Postulate** if we can show that the two triangles are congruent. Let's proceed with that.

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### Steps in the Proof:

1. **Statement**: \( \overline{DE} \cong \overline{FG} \)
   **Reason**: Given (the two sides are congruent)

2. **Statement**: \( \angle EDF \cong \angle GFD \)
   **Reason**: Given (the two angles are congruent)

3. **Statement**: \( \overline{EF} \cong \overline{GF} \) 
   **Reason**: Common side (both triangles share this side)

4. **Statement**: \( \triangle EFD \cong \triangle GDF \)
   **Reason**: ASA Congruence Postulate (Two angles and the included side are congruent)

5. **Statement**: \( \angle EFD \cong \angle GDF \)
   **Reason**: Corresponding angles of congruent triangles

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This completes the proof using the ASA postulate.

Let me know if you want more details on any step or if you have any questions!

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### Related Questions:
1. How does the ASA postulate prove triangle congruence?
2. Can other congruence postulates (e.g., SAS, SSS) be used in this proof?
3. Why is the third side not needed to prove congruence in this case?
4. How do we know that \(\overline{EF} \cong \overline{GF}\)?
5. What other geometric properties could help prove angle congruence?

### Tip:
Always look for shared sides or angles in diagrams, as they often lead to simpler proofs using congruence postulates.

Use the given information to prove that \( \angle EFD \cong \angle GDF \).

In this geometry problem, learn how to prove that \( \angle EFD \cong \angle GDF \) using the ASA Congruence Postulate. The solution involves analyzing the properties of congruent triangles and using key geometric theorems. Suitable for students in Grades 9-10.

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