https://r2.erweima.ai/mathbot/image/1724345573548bbaowjn4.jpg

To solve the problem, let's first analyze the given image.

### Problem Statement:
- You are asked to prove that \(\triangle ABC \cong \triangle DBE\).
- Given that \(\overline{AC} \parallel \overline{DE}\) and \(\overline{AB} = \overline{DB}\).

### Proof Structure:
The goal is to prove the congruence of two triangles, \(\triangle ABC\) and \(\triangle DBE\).

#### **Step 1: Identifying Given Information**
- **Parallel Lines:** \(\overline{AC} \parallel \overline{DE}\)
- **Equal Segments:** \(\overline{AB} = \overline{DB}\)

#### **Step 2: Congruence Criteria**
To prove that two triangles are congruent, we can use several criteria:
1. **SSS (Side-Side-Side)**
2. **SAS (Side-Angle-Side)**
3. **ASA (Angle-Side-Angle)**
4. **AAS (Angle-Angle-Side)**
5. **HL (Hypotenuse-Leg for Right Triangles)**

#### **Step 3: Apply Parallel Line Properties**
Since \(\overline{AC} \parallel \overline{DE}\) and \(\overline{AB} = \overline{DB}\), we can use the properties of parallel lines:
- **Alternate Interior Angles** are equal when a transversal cuts two parallel lines.

Let’s denote the angles:
- \(\angle CAB = \angle EDB\) (Alternate Interior Angles)
- \(\angle ABC = \angle DBE\) (They are vertically opposite angles)

#### **Step 4: Prove Congruence Using ASA**
Based on the given and derived information:
- \(\overline{AB} = \overline{DB}\) (Given)
- \(\angle CAB = \angle EDB\) (Alternate Interior Angles)
- \(\angle ABC = \angle DBE\) (Vertically Opposite Angles)

Thus, by the **ASA (Angle-Side-Angle) criterion**, \(\triangle ABC \cong \triangle DBE\).

### Conclusion:
The triangles \(\triangle ABC\) and \(\triangle DBE\) are congruent by the ASA criterion.

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Would you like a more detailed explanation of any step? 

Here are some related questions:
1. What are the different criteria for proving triangle congruence?
2. How do parallel lines affect the angles formed by a transversal?
3. What are alternate interior angles, and why are they equal?
4. What is the significance of vertically opposite angles in geometry?
5. Can you use another method to prove the congruence of \(\triangle ABC\) and \(\triangle DBE\)?

**Tip:** When working with parallel lines, always look out for equal corresponding and alternate interior angles to help with proofs.

Learn how to prove triangle congruence between triangles ABC and DBE using the ASA (Angle-Side-Angle) criterion. This problem involves parallel lines, alternate interior angles, and vertically opposite angles, essential in geometric proofs. Suitable for high school students.

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