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The problem presented in the image asks to prove that segment \( AB \cong CD \), given that \( AC \cong BD \). This involves showing that two line segments are congruent based on the information provided.

### Proof:
1. **Given**: \( AC \cong BD \) (as per the problem statement).
2. **To Prove**: \( AB \cong CD \).

### Step-by-Step Reasoning:
1. **Step 1**: 
   - **Statement**: \( AC \cong BD \).
   - **Reason**: This is given directly in the problem.

2. **Step 2**: 
   - **Statement**: Segment \( AB \cong CD \).
   - **Reason**: Since \( AC \cong BD \), and both \( AB \) and \( CD \) are parts of \( AC \) and \( BD \) respectively, the transitive property of congruence can be applied. Therefore, \( AB \) is congruent to \( CD \).

### Additional Explanation:
- The **Transitive Property of Congruence** states that if two segments are each congruent to a third segment, then they are congruent to each other.

Would you like more details on any part of this proof or have any further questions? Here are some related questions to consider:

1. What is the transitive property in geometry, and how is it used?
2. How can you prove segment congruence using the midpoint?
3. How is the segment addition postulate related to segment congruence?
4. What are other methods to prove two segments are congruent?
5. How does the concept of congruence apply to angles?

**Tip**: When proving congruence, always consider if there are properties like transitive or reflexive properties that could simplify the proof.

Prove AB ≅ CD, given that AC ≅ BD. This involves showing that two line segments are congruent based on the given information.

In this geometry problem, we prove that segment AB is congruent to CD using the given information AC ≅ BD. The solution applies the transitive property of congruence, showing how segments related by congruence can be proven equal. Ideal for students in Grades 7-10.

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