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Let's fill out the missing parts of the proof in Problem 7 from the uploaded document.

### Problem 7:
**Given:** \( E \) is the midpoint of \( \overline{AC} \), \( DE = EC \)  
**Prove:** \( DE \cong AE \)

### Proof:

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

2. **Statement:** \( AE = EC \)
   **Reason:** Definition of Midpoint

3. **Statement:** \( DE = EC \)
   **Reason:** Given

4. **Statement:** \( AE = DE \)
   **Reason:** Transitive Property of Equality (from statements 2 and 3)

5. **Statement:** \( DE \cong AE \)
   **Reason:** Definition of Congruence (congruence due to equality from statement 4)

6. **Statement:** \( DE \cong AE \)
   **Reason:** Restatement of what needs to be proven (already covered in step 5)

This completes the proof for Problem 7.

Do you need any details on this proof or have any other questions?

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### Follow-up Questions:
1. Would you like to see the complete proof for problem 8?
2. Need help understanding the Transitive Property of Equality used in the proof?
3. Are you looking for more examples of geometric proofs involving midpoints?
4. Do you want explanations on how to determine congruent segments in geometry?
5. Interested in learning about other properties of midpoints beyond congruence?

**Tip:** Always review the definitions and properties used in proofs, like the Definition of Midpoint and Transitive Property, as understanding these foundational elements will enhance your ability to solve more complex problems.

Given: E is the midpoint of AC, DE = EC. Prove: DE ≅ AE

Learn how to prove the congruence of line segments using the midpoint and transitive property of equality. This proof demonstrates key concepts in geometry such as midpoint, congruence, and the transitive property. Suitable for students in Grades 8-10.

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