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To complete the flowchart proof for proving that triangles \( \triangle ABE \) and \( \triangle CDE \) are congruent using the Side-Angle-Side (SAS) postulate, follow these steps:

### Given:
- \( E \) is the midpoint of segment \( AC \).

### Diagram Information:
The diagram shows two triangles, \( \triangle ABE \) and \( \triangle CDE \), with the following relationships:
1. \( E \) is the midpoint of \( AC \), implying \( AE = EC \).
2. \( \overline{BE} \cong \overline{DE} \), indicated by the markings.

### Proof Steps:

1. **First Statement and Reason:**
   - **Statement:** \( E \) is the midpoint of \( \overline{AC} \).
   - **Reason:** Given.

2. **Second Statement and Reason:**
   - **Statement:** \( \overline{AE} \cong \overline{EC} \).
   - **Reason:** Definition of a midpoint (a midpoint divides a segment into two congruent segments).

3. **Third Statement and Reason:**
   - **Statement:** \( \overline{BE} \cong \overline{DE} \).
   - **Reason:** Given (as indicated by the diagram).

4. **Fourth Statement and Reason:**
   - **Statement:** \( \angle ABE \cong \angle CDE \).
   - **Reason:** Vertical angles are congruent (since \( \angle ABE \) and \( \angle CDE \) are opposite each other).

5. **Final Statement and Reason (Provided):**
   - **Statement:** \( \triangle ABE \cong \triangle CDE \).
   - **Reason:** SAS (Side-Angle-Side) Congruence Postulate.

These steps establish that \( \triangle ABE \cong \triangle CDE \) through the SAS criterion: two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle.

Would you like more details or have any questions?

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Here are five related questions:
1. What other congruence criteria besides SAS can be used to prove triangle congruence?
2. How can we prove triangles congruent if we are given two angles and a non-included side?
3. What is the significance of vertical angles in proofs involving triangle congruence?
4. How would the proof change if we were given \( \overline{AB} \cong \overline{CD} \) instead of \( \overline{BE} \cong \overline{DE} \)?
5. Why is it important to provide reasons for each step in a geometric proof?

**Tip:** Always look for congruent segments or angles that can be inferred from given information and diagram markings. This can help in identifying the appropriate congruence postulate to use.

Given E is the midpoint of AC, complete the flowchart proof below. Note that the last statement and reason have been filled in for you.

Complete a flowchart proof involving two triangles with E as the midpoint of AC. Use the SAS postulate to establish congruence between triangle ABE and triangle CDE. Learn how geometric properties like midpoints and vertical angles support congruence in this detailed proof suitable for students in Grades 8-10.

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