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To solve this problem, you need to complete a two-column proof to show that \( x = 3 \) given the relationships among the line segments in the diagram.

### Given:
- \( AB = 3x - 5 \)
- \( BC = 4x \)
- \( AC = 16 \)

Since \( A \), \( B \), and \( C \) are collinear and \( AB + BC = AC \), we can set up an equation and solve for \( x \).

### Two-Column Proof:

| **Statements**                           | **Reasons**                                           |
|------------------------------------------|-------------------------------------------------------|
| 1. \( AB = 3x - 5 \), \( BC = 4x \), \( AC = 16 \) | 1. Given                                              |
| 2. \( AB + BC = AC \)                    | 2. Segment Addition Postulate                         |
| 3. \( (3x - 5) + 4x = 16 \)              | 3. Substitution of given values for \( AB \), \( BC \), and \( AC \) |
| 4. \( 7x - 5 = 16 \)                     | 4. Combine like terms                                 |
| 5. \( 7x = 21 \)                         | 5. Add 5 to both sides                                |
| 6. \( x = 3 \)                           | 6. Divide both sides by 7                             |
| 7. \( x = 3 \)                           | 7. Proved (Conclusion)                                |

This proof demonstrates that the value of \( x \) is 3, satisfying the conditions of the problem.

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Do you need further details or have any questions?

Here are five related questions to expand on this concept:
1. How can the segment addition postulate be applied to other geometric problems?
2. What is the process for solving equations involving multiple variables?
3. How can we use algebra to solve geometric problems involving line segments?
4. What other geometric postulates are commonly used in proofs?
5. How can we check if our solution for \( x \) is correct?

**Tip**: When solving geometry proofs, it's crucial to understand and apply postulates and theorems systematically to justify each step clearly.

Write a two-column proof for the following: Given AB = 3x - 5, BC = 4x, and AC = 16, prove x = 3.

Learn how to solve a geometric proof involving the segment addition postulate. Given AB = 3x - 5, BC = 4x, and AC = 16, follow the two-column proof to determine that x equals 3. Suitable for high school geometry students in Grades 9-10.

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