https://file.aiquickdraw.com/mathbot/file/1725687085874h2y0iqu9.jpg

The image you uploaded contains a geometric proof with a two-column format: one side for "Statements" and the other for "Reasons." The goal is to prove that segment **AC = BD** given the statements:

**Given:**
- \( AB = BC \)
- \( DE = CD \)

### Proof Layout:

| **Statement**                            | **Reason**       |
|------------------------------------------|------------------|
| 1. \( AB = BC \), \( DE = CD \)          | Given            |
| 2. \( AC = AB + BC \), \( BD = BC + CD \) | Segment Addition |
| 3. \( AC = BD \)                         | Substitution     |

### Explanation:

1. **Statement 1:** \( AB = BC \) and \( DE = CD \) are provided as the given information.
2. **Statement 2:** Applying the segment addition postulate, we express the lengths of \( AC \) and \( BD \) as sums of segments. Specifically:
   - \( AC = AB + BC \)
   - \( BD = BC + CD \)
3. **Statement 3:** Using the given information, we substitute the equalities into the second statement. Since \( AB = BC \) and \( CD = DE \), we conclude that \( AC = BD \).

Would you like any further details or a more in-depth explanation?

Here are 5 related questions to help expand your understanding:
1. How does the segment addition postulate work in this proof?
2. What role do the given equalities \( AB = BC \) and \( DE = CD \) play in reaching the conclusion?
3. Can this proof be applied to other types of geometric figures?
4. What is the importance of substitution in mathematical proofs?
5. How would you write the converse of this proof?

**Tip:** When working through geometric proofs, always ensure you understand each postulate or theorem being applied—this helps clarify the logic behind each step.

Given that AB = BC and DE = CD, prove that AC = BD using segment addition and substitution.

This geometric proof demonstrates how to prove AC = BD using the segment addition postulate and substitution. Given AB = BC and DE = CD, this problem is suitable for students in Grades 9-10 and covers important concepts such as segment addition and substitution.

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