Question

Payton is working on a proof that consecutive interior angles are supplementary. Given that   n ∥ m , fill in the following 2 column proof.(2 points)

Answer 1

To prove that consecutive interior angles are supplementary given that lines \( n \) and \( m \) are parallel, we can use a two-column proof format. Below is a completed version of the proof:

| **Statements** | **Reasons** |
|----------------------------------------------------------|------------------------------------------------|
| 1. Lines \( n \) and \( m \) are parallel. | 1. Given |
| 2. A transversal intersects lines \( n \) and \( m \). | 2. Definition of transversal |
| 3. Identify the consecutive interior angles (let's call them \( \angle 1 \) and \( \angle 2 \)). | 3. Definition of consecutive interior angles |
| 4. \( \angle 1 \) and \( \angle 2 \) are located on the same side of the transversal between lines \( n \) and \( m \). | 4. By the configuration of the diagram using the transversal |
| 5. Alternate interior angle theorem states that \( \angle 1 \) is equal to \( \angle 3 \) (where \( \angle 3 \) is an angle formed on the exterior of \( m \)). | 5. Corresponding angles are congruent (for parallel lines) |
| 6. \( \angle 2 \) and \( \angle 3 \) form a linear pair. | 6. Linear pair definition |
| 7. Therefore, \( \angle 1 + \angle 2 = 180^\circ \) (since the angles in a linear pair are supplementary). | 7. Linear Pair Postulate |
| 8. Thus, consecutive interior angles \( \angle 1 \) and \( \angle 2 \) are supplementary. | 8. Conclusion from previous statements |

This proof demonstrates that consecutive interior angles formed by a transversal intersecting two parallel lines are indeed supplementary.