https://file.aiquickdraw.com/mathbot/file/1729220167689jhdxkogh.jpg

In the problem shown, the goal is to prove that \( AB = CD \), given that \( AC = BD \).

Let's walk through the steps provided:

### Given: \( AC = BD \)

1. From the diagram, we know that:
   - \( AC = AB + BC \) (segment addition on the line \( A \) to \( C \)).
   - \( BD = BC + CD \) (segment addition on the line \( B \) to \( D \)).

2. The steps provided outline the reasoning:

   1. \( AC = BD \) (Given)
   2. \( AC = AB + BC \) (segment addition for \( AC \))
   3. \( BD = BC + CD \) (segment addition for \( BD \))
   4. Since \( AC = BD \), we substitute from steps 2 and 3 to get:
      \[
      AB + BC = BC + CD
      \]
   5. Subtracting \( BC \) from both sides gives:
      \[
      AB = CD
      \]

Thus, the proof is complete: \( AB = CD \).

### Would you like more details on any step, or do you have any questions?

Here are 5 questions that expand on the proof:
1. What is the definition of segment addition in geometry?
2. How does subtracting equal segments from both sides of an equation preserve equality?
3. Can this proof be applied to other geometric figures or only to line segments?
4. What are some real-world applications of segment addition and equality in geometry?
5. How would the proof change if the given was \( AC > BD \) instead of \( AC = BD \)?

### Tip:
Always check the conditions of equality and make sure all steps follow logically from the given information, especially when dealing with geometric proofs.

In this geometry proof, we show that AB = CD, given that AC = BD. The solution walks through the Segment Addition Theorem and uses logical steps to prove the equality of line segments. Suitable for students in Grades 6-8.

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