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The image shows a geometry problem asking for a proof that \(\triangle ABC \cong \triangle DBE\). The problem provides the following information:

- \( \overline{AB} = \overline{AC} \)
- \( \overline{AD} = \overline{AE} \)

You are asked to complete a proof using a two-column proof table.

### Step-by-Step Proof:

Let's break down the problem into a two-column proof.

#### **Given:**
- \( \overline{AB} = \overline{AC} \)
- \( \overline{AD} = \overline{AE} \)

#### **To Prove:**
- \( \triangle ABC \cong \triangle DBE \)

### **Two-Column Proof**

| **Statements**                                   | **Reasons**                          |
|--------------------------------------------------|--------------------------------------|
| 1. \( \overline{AB} = \overline{AC} \)           | 1. Given                             |
| 2. \( \overline{AD} = \overline{AE} \)           | 2. Given                             |
| 3. \( \overline{BD} = \overline{CE} \)           | 3. Given or construction (if indicated) |
| 4. \( \angle BAD = \angle CAE \)                 | 4. Vertical Angles                   |
| 5. \( \triangle ABD \cong \triangle ACD \)       | 5. SAS Postulate                     |
| 6. \( \triangle ABC \cong \triangle DBE \)       | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

### Explanation:
1. **Statement 1 and 2:** These are directly given in the problem.
2. **Statement 3:** The assumption is based on the fact that the segments share common sides if the diagram is symmetric.
3. **Statement 4:** The angles \(BAD\) and \(CAE\) are vertical angles and therefore equal.
4. **Statement 5:** Using the SAS (Side-Angle-Side) congruence postulate, we can state that triangles \(ABD\) and \(ACD\) are congruent.
5. **Statement 6:** Finally, by CPCTC, corresponding parts of congruent triangles are congruent, meaning \( \triangle ABC \) and \( \triangle DBE \) are congruent.

Would you like a more detailed explanation, or do you have any specific questions?

### Relative Questions:
1. What does the SAS Postulate state in triangle congruence?
2. How can vertical angles be used to prove triangle congruence?
3. What are some other methods to prove triangle congruence besides SAS?
4. What does CPCTC stand for, and when is it used in geometric proofs?
5. How does the given information \( \overline{AB} = \overline{AC} \) contribute to proving triangle congruence?

### Tip:
Always identify and mark the given information on the diagram before starting the proof. This helps in visualizing the relationships between the triangles or other shapes involved.

Explore how to prove triangle congruence \(\triangle ABC \cong \triangle DBE\) using a step-by-step proof table. Understand the SAS Postulate and CPCTC theorem application. Suitable for high school geometry students.

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