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We are given the following information from the problem:

- \( B \) is the midpoint of \( \overline{AC} \).
- \( E \) is the midpoint of \( \overline{AD} \).
- \( \angle 7 \cong \angle 8 \).
- \( \angle ECD \cong \angle BDC \).

### Objective:
Prove that \( \overline{AC} \cong \overline{AD} \).

### Approach:

1. **Given midpoints**:
   - Since \( B \) is the midpoint of \( \overline{AC} \), this implies that \( AB = BC \).
   - Since \( E \) is the midpoint of \( \overline{AD} \), this implies that \( AE = ED \).

2. **Congruence of angles**:
   - We are told that \( \angle 7 \cong \angle 8 \), and we also know that \( \angle ECD \cong \angle BDC \).

3. **Triangle analysis**:
   - Consider triangles \( \triangle ABE \) and \( \triangle BCD \).
   - By the given information, there are multiple congruent parts:
     - \( AB = BC \) because \( B \) is the midpoint of \( \overline{AC} \).
     - \( AE = ED \) because \( E \) is the midpoint of \( \overline{AD} \).
     - \( \angle AEB = \angle CED \) since both are vertical angles.
   
4. **Application of SAS (Side-Angle-Side)**:
   - Using the congruence of two sides and the included angle (SAS), we can conclude that \( \triangle ABE \cong \triangle CED \).
   
5. **Final Conclusion**:
   - Since \( \triangle ABE \cong \triangle CED \), it follows that \( \overline{AC} \cong \overline{AD} \).

This completes the proof.

Would you like further details on any part of the explanation? Here are five related questions:

1. Why does proving the triangles are congruent imply that \( \overline{AC} \cong \overline{AD} \)?
2. How does the midpoint property help in this geometric proof?
3. Can you explain why vertical angles are congruent?
4. What is the significance of using the SAS (Side-Angle-Side) congruence criterion here?
5. Could the proof work with a different congruence postulate, like ASA or SSS?

### Tip:
When proving triangles congruent, always ensure you identify at least two congruent sides and the included angle, or use other congruence postulates like ASA, SSS, or AAS for an accurate proof.

Given: B is the midpoint of AC. E is the midpoint of AD. ∠7 ≅ ∠8. ∠ECD ≅ ∠BDC. Prove: AC ≅ AD.

Learn how to prove AC ≅ AD by using triangle congruence, midpoints, and the SAS criterion in this geometry problem. Detailed step-by-step solution with core concepts and theorems for students in Grades 9-10.

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