Question

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

Explanation:

Step1: Recall property of parallel lines

When two parallel lines \(x\parallel y\) are cut by a transversal, same - side interior angles are supplementary. \(\angle3\) and \(\angle 8\) are same - side interior angles. So, \(m\angle3 + m\angle8=180^{\circ}\) because of the Same - Side Interior Angles Postulate, not the Transitive Property of Equality.

Step2: Note vertical - angle relationship

\(\angle5\) and \(\angle8\) are vertical angles. By the Vertical Angles Theorem, \(m\angle5=m\angle8\).

Step3: Substitute

Since \(m\angle3 + m\angle8 = 180^{\circ}\) and \(m\angle5=m\angle8\), we can substitute \(m\angle5\) for \(m\angle8\) in the first equation: \(m\angle3+m\angle5 = 180^{\circ}\). Also, since \(m\angle3 + m\angle8=180^{\circ}\) and \(m\angle5=m\angle8\), we can say \(\angle3\cong\angle5\) because if two angles are supplementary to the same angle (or congruent angles), then they are congruent.

Answer:

The reason for \(m\angle3 + m\angle8 = 180\) should be "Same - Side Interior Angles Postulate". Then, since \(\angle5\) and \(\angle8\) are vertical angles (\(m\angle5 = m\angle8\) by Vertical Angles Theorem) and \(m\angle3+m\angle8 = 180^{\circ}\), we can conclude \(\angle3\cong\angle5\) (angles supplementary to congruent angles are congruent).