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.
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