https://file.aiquickdraw.com/mathbot/file/1729720862074lc7gnw6v.jpg

To solve this proof, we are asked to prove that \( AG \cong EC \). 

Here is a breakdown of the problem based on the given image:

### Given Information:
- \( \triangle AED \) and \( \triangle CGF \) are both right triangles because \( AE \perp AD \) and \( CG \perp CF \), meaning angles \( \angle AED \) and \( \angle CGF \) are right angles.
- \( DE \cong FG \) and \( DA \cong FC \) are also given.

### Strategy:
1. **Triangles Identification**: The triangles \( \triangle AED \) and \( \triangle CGF \) are right triangles, and we know the relationships \( DE \cong FG \) and \( DA \cong FC \).
   
2. **Congruence Reasoning**: The key here is to use the information about the sides of these right triangles to show that the corresponding parts of the two triangles (specifically \( AG \) and \( EC \)) are congruent.

3. **Hypotenuse-Leg Theorem (HL)**: This theorem is a special case for proving that two right triangles are congruent. It states that if two right triangles have a congruent hypotenuse and one congruent leg, then the triangles are congruent.

### Steps in the Proof:

#### Step 1: 
- We have two right triangles: \( \triangle AED \) and \( \triangle CGF \). Given that \( \angle AED \cong \angle CGF \) (both are 90°), we can use the HL Theorem.

#### Step 2: 
- The hypotenuse of \( \triangle AED \) is \( DA \), and for \( \triangle CGF \), it is \( FC \). We are given that \( DA \cong FC \).
- Additionally, one pair of legs, \( DE \) and \( FG \), are also congruent.

#### Step 3: 
- By the **Hypotenuse-Leg Theorem**, we conclude that \( \triangle AED \cong \triangle CGF \).

#### Step 4: 
- Since the triangles are congruent, the corresponding parts of the triangles are congruent. Therefore, we have \( AG \cong EC \), as required.

### Conclusion:
We have successfully shown that \( AG \cong EC \) by using the Hypotenuse-Leg Theorem and the given congruencies of sides and angles.

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Do you need further clarification or any specific details about a particular step in this proof?

### Questions to Explore:
1. What is the Hypotenuse-Leg Theorem, and when is it applicable?
2. How do right triangles play a role in proving congruence?
3. Why does congruence of sides matter in geometric proofs?
4. Could we prove this congruence using another method, such as Side-Angle-Side (SAS)?
5. What is the significance of corresponding parts of congruent triangles (CPCTC)?

**Tip**: When proving triangle congruence, always look for right triangles, as they allow for specific theorems like HL to simplify the process.

This problem demonstrates how to prove that AG ≅ EC by using the Hypotenuse-Leg (HL) Theorem. It involves geometric concepts related to right triangles and congruent sides. Suitable for students in Grades 9-10.

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