Question

1. construct arguments prove the consecutive interior angles theorem. given: p || q prove: ∠1 and ∠3 are supplementary.

Explanation:

Step1: Recall corresponding - angles property

Since \(p\parallel q\), \(\angle1\) and \(\angle2\) are corresponding angles. So, \(\angle1=\angle2\) (Corresponding - Angles Postulate).

Step2: Note linear - pair relationship

\(\angle2\) and \(\angle3\) form a linear pair. By the Linear - Pair Postulate, \(\angle2+\angle3 = 180^{\circ}\).

Step3: Substitute \(\angle1\) for \(\angle2\)

Since \(\angle1=\angle2\), we can substitute \(\angle1\) into the equation \(\angle2+\angle3 = 180^{\circ}\). We get \(\angle1+\angle3 = 180^{\circ}\).

Answer:

Since \(\angle1+\angle3 = 180^{\circ}\), \(\angle1\) and \(\angle3\) are supplementary.