https://file.aiquickdraw.com/mathbot/file/172939026252193ihu52m.jpg

Let’s walk through the given problem from the image.

### **Given:**
1. \( \triangle AEC, \ \triangle BED, \ \triangle GEF \)
2. \( AE \cong CE \) and \( FE \cong GE \)

### **Prove:**
- \( \triangle DEC \cong \triangle BEA \)

### **Solution Approach:**

To prove the congruence \( \triangle DEC \cong \triangle BEA \), we need to find corresponding parts that are congruent. Let’s use the **Side-Angle-Side (SAS) Congruence** or other relevant theorems like **ASA**.

#### Step 1: Find common sides and angles.
1. **Shared Side**: Both triangles \( \triangle DEC \) and \( \triangle BEA \) share the segment \( DE = BE \).
2. **Congruent Sides**:  
   - \( AE \cong CE \) (Given).
   - \( FE \cong GE \) (Given).  
   Therefore, \( BE = GE \) as a result of segment relations in the larger figure.

#### Step 2: Prove angle congruence.
- Both triangles share the angle \( \angle AEB \), which corresponds to \( \angle DEC \) through vertical angles.
  
#### Step 3: Apply Congruence Criteria.
- By **ASA Congruence**:
  - \( AE \cong CE \) (given).
  - \( \angle AEB \cong \angle DEC \) (vertical angles).
  - \( BE \cong DE \) (equal segments).

Thus, \( \triangle DEC \cong \triangle BEA \) by ASA congruence.

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Do you want me to elaborate on any step? Or would you like further details on other congruence theorems?

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### 5 Related Questions:
1. What is the ASA congruence theorem, and how is it applied?
2. How do vertical angles help in proving congruence?
3. What is the difference between SAS and ASA congruence?
4. Can we apply any other congruence criteria in this problem (like SSS)?
5. How does segment congruence affect triangle congruence proofs?

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**Tip:** Always look for shared or common angles and sides in diagrams. These are usually key to proving triangle congruence efficiently.

Given the triangles AEC, BED, and GEF, with AE ≅ CE and FE ≅ GE, prove that triangle DEC ≅ triangle BEA.

Learn how to prove triangle congruence using the ASA Congruence Theorem and vertical angles. This problem involves triangles AEC, BED, and GEF, where AE ≅ CE and FE ≅ GE. Step-by-step geometric proof suitable for students in Grades 9-11.

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