https://file.aiquickdraw.com/mathbot/file/17270929333966mvz0zj6.jpg

To help complete this geometric proof, let's first understand the given diagram and steps:

### Given:
- \( K \) is the midpoint of \( JL \).
- \( M \) is the midpoint of \( LN \).
- \( JK = MN \).

### Statements:
The proof asks to show that \( KL \cong LM \).

### Statements to complete and their reasoning:

1. **Statement:** \( K \) is the midpoint of \( JL \), \( M \) is the midpoint of \( LN \).
   - **Reason:** Given.

2. **Statement:** \( JK = KL \), \( LM = MN \).
   - **Reason:** Definition of midpoint (Each midpoint divides the segment into two equal parts).

3. **Statement:** \( JK = MN \).
   - **Reason:** Given.

4. **Statement:** \( MN = KL \), \( LM = MN \).
   - **Reason:** Transitive property of equality (If \( JK = MN \) and \( JK = KL \), then \( MN = KL \)).

5. **Statement:** \( LM = KL \).
   - **Reason:** Symmetric property of equality (If \( MN = KL \), then \( LM = KL \)).

6. **Statement:** \( KL = LM \).
   - **Reason:** Reflexive property of equality.

7. **Conclusion:** \( KL \cong LM \).
   - **Reason:** Definition of congruent segments (Equal segments are congruent).

This completes the proof that \( KL \cong LM \).

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### Do you want more details or have any questions about this solution?

Here are five related questions:
1. What is the transitive property of equality in geometry?
2. How does the midpoint definition help in segment congruence proofs?
3. Can the symmetric property be used in proofs other than for segment lengths?
4. What is the difference between equality and congruence in geometry?
5. How do the midpoint properties apply in triangles?

**Tip:** When working with geometric proofs, always start by identifying the given information and using properties like midpoint, transitive, and symmetric properties to establish equalities or congruencies.

Complete the proof. Given K is the midpoint of JL, M is the midpoint of LN, JK = MN. Prove KL ≅ LM.

Complete a geometric proof to show that KL is congruent to LM given that K and M are midpoints. This problem covers midpoints, segment congruence, and properties of equality in geometry, suitable for Grades 8-10.

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