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### Given:
- \( \overline{AB} \cong \overline{CD} \)

### To Prove:
- \( \overline{AC} \cong \overline{BD} \)

### Proof:

| **Statements**                          | **Reasons**                     |
|-----------------------------------------|---------------------------------|
| 1. \( \overline{AB} \cong \overline{CD} \) | 1. Given                        |
| 2. \( AB = CD \)                        | 2. Definition of congruent segments |
| 3. \( AC + CD = AD \)                   | 3. Segment Addition Postulate    |
| 4. \( AB + BD = AD \)                   | 4. Segment Addition Postulate    |
| 5. \( CD + BD = AD \)                   | 5. Substitution (from step 2)    |
| 6. \( AC + CD = CD + BD \)              | 6. Substitution                 |
| 7. \( AC = BD \)                        | 7. Subtraction Property of Equality |
| 8. \( \overline{AC} \cong \overline{BD} \) | 8. Definition of congruent segments |

### Explanation:

1. The first step uses the **Given** information that \( \overline{AB} \cong \overline{CD} \).
2. By the **Definition of congruence**, congruent segments are equal in length, so \( AB = CD \).
3. The **Segment Addition Postulate** states that if you have three collinear points, then the length of one segment plus the length of the adjacent segment equals the total length of the combined segment. Thus, \( AC + CD = AD \).
4. The same postulate applies for the other segment, giving \( AB + BD = AD \).
5. From step 2, we know \( AB = CD \), so we can substitute \( CD \) for \( AB \) in the previous equation to get \( CD + BD = AD \).
6. By substitution, \( AC + CD = CD + BD \).
7. Using the **Subtraction Property of Equality**, subtract \( CD \) from both sides, yielding \( AC = BD \).
8. Finally, by the **Definition of congruent segments**, if two segments have the same length, they are congruent, so \( \overline{AC} \cong \overline{BD} \).

Would you like further clarification on any of the steps, or more details? 

Here are five related questions for practice:
1. What is the Segment Addition Postulate and when do we use it?
2. How does the Substitution Property work in geometry proofs?
3. What is the difference between segment equality and segment congruence?
4. Why do we need the Definition of Congruence in the final step?
5. Can this proof be extended to any non-collinear points? Why or why not?

**Tip:** Always write out each substitution clearly to avoid confusion when transitioning between steps!

This step-by-step geometry proof demonstrates how to prove that two segments, AC and BD, are congruent using the Segment Addition Postulate and the Substitution Property. The proof is suitable for students in Grades 8-10 learning about congruent segments and geometric reasoning.

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