https://file.aiquickdraw.com/mathbot/file/17295601100617o87459f.jpg

To complete this flowchart proof, let's break down the key components of the proof.

**Given:**
\( \overline{AB} \parallel \overline{DC} \)

The goal is to prove \( \triangle ABE \cong \triangle CDE \) using the Angle-Angle-Side (AAS) congruence criterion. The final statement has already been provided for us, so we need to fill in the missing steps leading to that conclusion.

### Step 1: Use of Parallel Lines
Since \( \overline{AB} \parallel \overline{DC} \), we can apply the property of alternate interior angles.

**Statement:**  
\( \angle AEB \cong \angle CED \)

**Reason:**  
Alternate Interior Angles Theorem (because the lines are parallel).

### Step 2: Use of Vertical Angles
Notice that \( \angle AEB \) and \( \angle CED \) share the vertex \( E \), which means we can apply the property of vertical angles.

**Statement:**  
\( \angle BAE \cong \angle DCE \)

**Reason:**  
Vertical Angles are Congruent.

### Step 3: Use of Congruent Sides
Given the markings on the diagram, \( \overline{AE} \cong \overline{CE} \), which is a third congruent element needed for the AAS congruence theorem.

**Statement:**  
\( \overline{AE} \cong \overline{CE} \)

**Reason:**  
Given (as indicated by the marks on the diagram).

### Conclusion:
We can now use AAS (Angle-Angle-Side) to conclude that \( \triangle ABE \cong \triangle CDE \).

- **Final Statement:** \( \triangle ABE \cong \triangle CDE \)
- **Reason:** AAS (Angle-Angle-Side).

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This completes the proof. Let me know if you want more details or have any questions! Here are some related questions for further exploration:

1. How do parallel lines lead to the congruence of alternate interior angles?
2. What other congruence theorems can be used to prove triangles congruent?
3. How does the vertical angles theorem work in geometric proofs?
4. What other cases require the AAS (Angle-Angle-Side) theorem in triangle congruence?
5. How can we differentiate between AAS and ASA (Angle-Side-Angle)?

**Tip:** Always start by marking all given information on the diagram. This helps to visualize relationships like congruent angles and sides early in the proof.

Given that AB || DC, complete the flowchart proof. The goal is to prove that ΔABE ≅ ΔCDE using the AAS (Angle-Angle-Side) congruence criterion. The last statement and reason have already been provided.

Solve a flowchart proof where AB || DC, proving that ΔABE ≅ ΔCDE using the AAS (Angle-Angle-Side) theorem. This problem explores key geometry concepts, including parallel lines, alternate interior angles, and triangle congruence. Suitable for students in Grades 9-11.

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