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It looks like you've uploaded an image with a proof diagram. The problem appears to be about completing the steps of a geometric proof, where we have statements on the left and reasons (justifications) on the right.

Let's go step by step to solve it.

### Proof Layout:

**Given:**
- \( SU \cong LR \) (Segment \( SU \) is congruent to segment \( LR \))
- \( TU \cong LN \) (Segment \( TU \) is congruent to segment \( LN \))

The goal is to prove some equality involving segment sums.

### Analyzing the proof:

1. **Statement 1:**
   \( SU \cong LR, TU \cong LN \)  
   **Reason 1:** This is **Given**.

2. **Statement 2:**  
   \( SU = LR \), \( TU = LN \)  
   **Reason 2:** This follows from the **definition of congruent segments**. Congruent segments have equal lengths.

3. **Statement 3:**  
   \( SU = ST + TU \) and \( LR = LN + NR \)  
   **Reason 3:** This uses the **Segment Addition Postulate**: The length of a segment is the sum of its parts.

4. **Statement 4:**  
   \( ST + TU = LN + NR \)  
   **Reason 4:** We used the equalities in previous steps and substituted the values for \( SU \) and \( LR \) from Statement 3. This step is a result of **Substitution**.

5. **Statement 5:**  
   \( ST + TU = LN + NR \)  
   **Reason 5:** This is simply a repetition of the previous statement to remind us that both sides are equal. No new step here.

6. **Statement 6:**  
   \( ST + LN = LN + NR - LN \)  
   **Reason 6:** This is simplifying the equation by subtracting \( LN \) from both sides. This uses the **Subtraction Property of Equality**.

7. **Statement 7:**  
   \( ST + LN = NR \)  
   **Reason 7:** After subtracting, the equation is simplified to this. This uses the **Substitution Property of Equality**.

8. **Final Conclusion:**  
   \( ST + LN = NR \)  
   **Reason 8:** This is the **Final Conclusion** from the previous steps.

### Complete proof:

| Statements                                    | Reasons                                  |
|-----------------------------------------------|------------------------------------------|
| 1. \( SU \cong LR, TU \cong LN \)             | 1. Given                                 |
| 2. \( SU = LR, TU = LN \)                     | 2. Definition of congruent segments       |
| 3. \( SU = ST + TU, LR = LN + NR \)           | 3. Segment Addition Postulate            |
| 4. \( ST + TU = LN + NR \)                    | 4. Substitution                          |
| 5. \( ST + TU = LN + NR \)                    | 5. Rewriting equality                    |
| 6. \( ST + LN = LN + NR - LN \)               | 6. Subtraction Property of Equality       |
| 7. \( ST + LN = NR \)                         | 7. Substitution Property of Equality      |
| 8. Final conclusion: \( ST + LN = NR \)       | 8. Conclusion                            |

Would you like further details or clarifications on any step?

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Here are five related questions to expand on this topic:

1. What is the Segment Addition Postulate, and how is it used in geometric proofs?
2. Can you explain the Substitution Property of Equality with another example?
3. How do you know when to apply the Subtraction Property of Equality in a proof?
4. What is the difference between a theorem and a postulate in geometry?
5. How do congruent segments influence the structure of a geometric proof?

**Tip:** When solving geometric proofs, make sure to list all given information and clearly write down what you need to prove. This helps in choosing the right properties and postulates to apply.

Complete the geometric proof involving congruent segments SU, LR, TU, and LN.

Step-by-step solution for a geometric proof involving congruent segments SU, LR, TU, and LN. The proof uses the Segment Addition Postulate, Substitution, and Subtraction Properties of Equality. Suitable for students in Grades 9-10.

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