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The image contains three geometric proofs where the goal is to prove specific segment congruences based on given information about midpoints or congruent segments. Let’s go through the first problem in detail and outline the proofs one by one.

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### Proof 1 (Problem 4):

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

**Prove:**
- \( KL \cong LM \).

**Proof Outline:**
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 (since midpoints divide the segment into two equal halves).
   
3. **Statement:** \( JK = MN \).  
   **Reason:** Given.

4. **Statement:** \( MN = KL \), \( LM = MN \).  
   **Reason:** Transitive property of equality (from steps 2 and 3).

5. **Statement:** \( LM = KL \).  
   **Reason:** Substitution.

6. **Statement:** \( KL \cong LM \).  
   **Reason:** Segments are congruent if their lengths are equal.

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### Proof 2 (Problem 5):

**Given:**
- \( XY \cong UV \), \( YZ \cong TU \).

**Prove:**
- \( XZ \cong TV \).

**Proof Outline:**
1. **Statement:** \( XY \cong UV \), \( YZ \cong TU \).  
   **Reason:** Given.

2. **Statement:** \( XY = UV \), \( YZ = TU \).  
   **Reason:** Definition of congruence.

3. **Statement:** \( XY + YZ = XZ \), \( UV + TU = TV \).  
   **Reason:** Segment addition postulate.

4. **Statement:** \( UV + YZ = XZ \), \( YZ + UV = TV \).  
   **Reason:** Substitution from step 2.

5. **Statement:** \( XZ = TV \).  
   **Reason:** Transitive property.

6. **Statement:** \( XZ \cong TV \).  
   **Reason:** Definition of congruence (since their lengths are equal).

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### Proof 3 (Problem 6):

**Given:**
- \( WY \cong YZ \), \( XY \cong VY \).

**Prove:**
- \( XZ \cong VW \).

**Proof Outline:**
1. **Statement:** \( WY \cong YZ \), \( XY \cong VY \).  
   **Reason:** Given.

2. **Statement:** \( WY = YZ \), \( XY = VY \).  
   **Reason:** Definition of congruence.

3. **Statement:** \( XY + YZ = XZ \).  
   **Reason:** Segment addition postulate.

4. **Statement:** \( VY + YW = VW \).  
   **Reason:** Segment addition postulate.

5. **Statement:** \( XZ = VW \).  
   **Reason:** Transitive property of equality (steps 2, 3, and 4).

6. **Statement:** \( XZ \cong VW \).  
   **Reason:** Definition of congruence (since their lengths are equal).

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Would you like further explanation on any of these proofs? Here are five additional questions related to this topic:

1. How does the transitive property apply in segment congruence?
2. What is the importance of the midpoint in segment equality?
3. Can the segment addition postulate be used in other geometric proofs?
4. How does congruence differ from equality in geometry?
5. What are some common properties used in geometric proofs?

**Tip:** Always remember that in geometric proofs, segment congruence can be established by showing the lengths are equal, even if congruence is not directly given.

Prove segment congruence based on given midpoints and segment relationships.

Learn how to prove segment congruence using midpoints, the segment addition postulate, and transitive properties. These problems focus on applying the midpoint theorem and segment congruence relationships, suitable for high school geometry students.

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