Question

complete the two - column proof. given: x||y prove: ∠3≅∠5 statements reasons 1) x||y 1) given 2) m∠3 + m∠8 = 180 2)

Explanation:

Step1: Identify angle - pair relationship

Since \(x\parallel y\), \(\angle3\) and \(\angle8\) are same - side interior angles. The same - side interior angles theorem states that when two parallel lines are cut by a transversal, same - side interior angles are supplementary.

Step2: Use angle - pair relationship to prove \(\angle3\cong\angle5\)

We know that \(\angle5\) and \(\angle8\) are vertical angles, so \(m\angle5 = m\angle8\) (vertical angles are congruent). From \(m\angle3+m\angle8 = 180\) (same - side interior angles are supplementary) and \(m\angle5 = m\angle8\), we can substitute \(\angle8\) with \(\angle5\) to get \(m\angle3+m\angle5 = 180\). Also, since \(x\parallel y\), \(\angle3\) and \(\angle5\) are alternate interior angles. Alternate interior angles are congruent when two parallel lines are cut by a transversal.

Answer:

1. Same - side interior angles are supplementary.