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I see you're working on a geometry problem that asks to write a two-column proof to show that ∠4 is supplementary to ∠5, given the conditions:

- \( a \parallel b \parallel y \)
- \( x \perp y \)

Let’s walk through how to set up the proof:

### Given Information:
- \( a \parallel b \parallel y \) (three parallel lines)
- \( x \perp y \) (line \( x \) is perpendicular to \( y \))

### To Prove:
- \( \angle 4 \) is supplementary to \( \angle 5 \)

### Diagram Analysis:
- Since \( a \parallel b \parallel y \) and \( x \perp y \), \( x \) is a transversal that forms right angles where it intersects \( y \), including \( \angle 5 \).
- You also know that \( \angle 4 \) is adjacent to \( \angle 5 \) and they form a linear pair.

### Two-Column Proof:

| **Statements**                                 | **Reasons**                                         |
|------------------------------------------------|-----------------------------------------------------|
| 1. \( a \parallel b \parallel y \)             | Given                                               |
| 2. \( x \perp y \)                             | Given                                               |
| 3. \( \angle 5 = 90^\circ \)                   | Definition of perpendicular lines                   |
| 4. \( \angle 4 + \angle 5 = 180^\circ \)       | Linear pair postulate                               |
| 5. \( \angle 4 \) is supplementary to \( \angle 5 \) | Definition of supplementary angles (\( \angle 4 + \angle 5 = 180^\circ \)) |

This proves that \( \angle 4 \) is supplementary to \( \angle 5 \).

Would you like further clarification or more details on any step?

Here are 5 questions to expand on this topic:
1. Why are \( \angle 4 \) and \( \angle 5 \) considered a linear pair?
2. How do you identify supplementary angles in parallel line transversals?
3. Can two right angles be supplementary?
4. What other angle relationships can be found in the diagram?
5. How does the perpendicularity of lines help in geometry proofs?

**Tip:** Always remember that linear pairs of angles formed by a transversal with parallel lines sum up to 180°, making them supplementary.

Given: a || b, x ⊥ y. Prove: ∠4 is supplementary to ∠5.

This geometry problem proves that ∠4 is supplementary to ∠5, given parallel lines and a perpendicular transversal. The solution explores concepts such as linear pairs and the definition of supplementary angles. Suitable for high school students in Grades 9-10.

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