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Can you help explain what the reasons should be?

To solve this problem, we need to provide reasons for each step in the proof table to show that \( AB = CD \) given that \( AC = BD \). Here's how we can complete the table with the correct reasoning:

### Proof Completion:

1. **Statement**: \( AC = AB + BC \)  
   **Reason**: Segment Addition Postulate.  
   *Explanation*: The Segment Addition Postulate states that if point \( B \) is between points \( A \) and \( C \), then \( AB + BC = AC \).

2. **Statement**: \( BD = BC + CD \)  
   **Reason**: Segment Addition Postulate.  
   *Explanation*: Similarly, the Segment Addition Postulate states that if point \( C \) is between points \( B \) and \( D \), then \( BC + CD = BD \).

3. **Statement**: \( AC = BD \)  
   **Reason**: Given.  
   *Explanation*: This is directly given in the problem statement.

4. **Statement**: \( AB + BC = BC + CD \)  
   **Reason**: Substitution.  
   *Explanation*: Since \( AC = AB + BC \) and \( BD = BC + CD \), and \( AC = BD \), we substitute \( AC \) with \( AB + BC \) and \( BD \) with \( BC + CD \) to get \( AB + BC = BC + CD \).

5. **Statement**: \( AB = CD \)  
   **Reason**: Subtraction Property of Equality.  
   *Explanation*: If we subtract \( BC \) from both sides of the equation \( AB + BC = BC + CD \), we get \( AB = CD \).

### Final Proof Table:

| Statement               | Reason                                  |
|-------------------------|------------------------------------------|
| 1. \( AC = AB + BC \)   | Segment Addition Postulate               |
| 2. \( BD = BC + CD \)   | Segment Addition Postulate               |
| 3. \( AC = BD \)        | Given                                    |
| 4. \( AB + BC = BC + CD \) | Substitution                        |
| 5. \( AB = CD \)        | Subtraction Property of Equality         |

Would you like a more detailed explanation or have any questions? Here are some additional questions that could help reinforce understanding:

1. What is the Segment Addition Postulate, and how is it applied in geometry?
2. How does substitution work in mathematical proofs?
3. Why is the Subtraction Property of Equality important in algebra?
4. Can you provide another example where substitution is used in a geometric proof?
5. How would the proof change if the given information were different?

**Tip:** In geometry proofs, always remember to carefully justify each step with a clear and logical reason.

Learn how to complete a geometry proof by providing logical reasons for each step to show that AB equals CD given that AC equals BD. This proof involves the Segment Addition Postulate and the Subtraction Property of Equality, suitable for students in Grades 8-10.

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