https://file.aiquickdraw.com/mathbot/file/1727617639987u7x01ymo.jpg

The image you uploaded shows a geometric proof where we need to prove that \( QR = ST \) using the given information:

- \( PR = SU \)
- \( PQ = TU \)

### Let's break down the steps in the proof:

1. **Statement:** \( PR = PQ + QR \)  
   **Reason:** This comes from the **segment addition postulate**, which states that the whole segment is the sum of its parts.

2. **Statement:** \( SU = ST + TU \)  
   **Reason:** Similarly, this is from the **segment addition postulate** applied to segment \( SU \).

3. **Statement:** \( PR = SU \)  
   **Reason:** This is **given** in the problem.

4. **Statement:** \( PQ + QR = ST + TU \)  
   **Reason:** Substitution. Since \( PR = SU \) from the given information, we can substitute \( PR \) and \( SU \) into the equations derived from the segment addition postulate.

5. **Statement:** \( PQ = TU \)  
   **Reason:** This is also **given** in the problem.

6. **Statement:** \( TU + QR = ST + TU \)  
   **Reason:** Substitution. From the previous steps, we substitute \( PQ = TU \) into the equation.

7. **Statement:** \( QR = ST \)  
   **Reason:** **Subtraction property of equality**. After subtracting \( TU \) from both sides, we arrive at the final result, \( QR = ST \).

### Here's a completed version of the table:

| Statement               | Reason                            |
|-------------------------|-----------------------------------|
| 1. \( PR = PQ + QR \)    | Segment Addition Postulate        |
| 2. \( SU = ST + TU \)    | Segment Addition Postulate        |
| 3. \( PR = SU \)         | Given                             |
| 4. \( PQ + QR = ST + TU \)| Substitution                     |
| 5. \( PQ = TU \)         | Given                             |
| 6. \( TU + QR = ST + TU \)| Substitution                     |
| 7. \( QR = ST \)         | Subtraction Property of Equality  |

Would you like more details or have any further questions?

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Here are 5 related questions for deeper understanding:

1. What is the segment addition postulate, and how does it apply in geometry?
2. How does the substitution property help in geometric proofs?
3. What is the subtraction property of equality, and why is it important here?
4. Can you think of another geometric situation where segment addition could be used?
5. How do properties of equality assist in manipulating equations in proofs?

**Tip:** In geometry, the key to many proofs lies in recognizing congruent segments or angles and using properties of equality to manipulate these relationships effectively.

Complete the proof by choosing the correct reason for each step in a geometric proof involving segments PR, SU, PQ, TU, and QR. Given: PR = SU and PQ = TU, Prove: QR = ST.

Learn how to complete a geometric proof showing QR = ST using given segment lengths and properties of equality. This example uses the segment addition postulate and the substitution property, ideal for Grades 8-10.

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