https://file.aiquickdraw.com/mathbot/file/1727216934772d5yt6iu6.jpg

From the image, I can see a two-column proof (labeled "Proof 3") where the objective is to prove \( AC = BD \) given that \( AB = CD \). Let's walk through the proof step by step based on the information available.

### **Proof Explanation**:

**Given:**
\( AB = CD \)

**To Prove:**
\( AC = BD \)

1. **Statement:** \( AB = CD \)
   - **Reason:** Given (this is the problem's initial condition)

2. **Statement:** \( BC = BC \)
   - **Reason:** Reflexive Property of Equality (any segment is equal to itself)

3. **Statement:** \( AB + BC = BC + CD \)
   - **Reason:** Segment Addition Postulate (if two segments are adjacent, their lengths can be added)

4. **Statement:** \( AC = BD \)
   - **Reason:** Substitution Property of Equality (since \( AB = CD \), we can substitute and simplify to get \( AC = BD \))

This concludes the proof, showing that the segments \( AC \) and \( BD \) are indeed equal.

Would you like further details on any of the steps or the logic behind the properties used?

Here are five related questions:
1. What is the Reflexive Property of Equality and how is it used in proofs?
2. Can you explain the Segment Addition Postulate with another example?
3. What other properties of equality are often used in geometry proofs?
4. How can the Substitution Property of Equality simplify expressions or equations?
5. How does this proof change if different segment relationships were given?

**Tip:** In geometry, always ensure the logical flow between statements is backed by a valid reason to strengthen your proof.

In this geometry proof, learn how to prove AC = BD given that AB = CD using the Segment Addition Postulate, Reflexive Property of Equality, and Substitution Property. This detailed proof is ideal for students in Grades 8-10.

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