Question

use the diagram to explain how you could use the consecutive interior angles theorem and a linear pair of angles to prove the alternate interior angles theorem. which of the following gives the correct explanation? by the consecutive angles theorem, m∠4 + m∠5 = 60°. as a linear pair, m∠4 + m∠3 = 90°. therefore, m∠4 + m∠3 = m∠4 + m∠5, so m∠3 = m∠5. by the consecutive angles theorem, m∠4 + m∠5 = 360°. as a linear pair, m∠4 + m∠3 = 180°. therefore, m∠4 + m∠3 = m∠4 + m∠5, so m∠3 = m∠6. by the consecutive angles theorem, m∠4 + m∠5 = 90°. as a linear pair, m∠4 + m∠3 = 180°. therefore, m∠4 + m∠3 = m∠4 + m∠5, so m∠3 = m∠4. by the consecutive angles theorem, m∠4 + m∠5 = 180°. as a linear pair, m∠4 + m∠3 = 180°. therefore, m∠4 + m∠3 = m∠4 + m∠5, so m∠3 = m∠5

Explanation:

Step1: Recall Consecutive Interior Angles Theorem

The Consecutive Interior Angles Theorem states that when two parallel lines are cut by a transversal, consecutive - interior angles are supplementary, so \(m\angle4 + m\angle5=180^{\circ}\).

Step2: Recall linear - pair property

A linear pair of angles is supplementary. So for the linear pair \(\angle4\) and \(\angle3\), \(m\angle4 + m\angle3 = 180^{\circ}\).

Step3: Use the transitive property

Since \(m\angle4 + m\angle3=180^{\circ}\) and \(m\angle4 + m\angle5 = 180^{\circ}\), by the transitive property of equality, \(m\angle4 + m\angle3=m\angle4 + m\angle5\). Subtracting \(m\angle4\) from both sides of the equation gives \(m\angle3=m\angle5\), which is part of the Alternate Interior Angles Theorem.

Answer:

By the Consecutive Angles Theorem, \(m\angle4 + m\angle5 = 180^{\circ}\). As a linear pair, \(m\angle4 + m\angle3 = 180^{\circ}\). Therefore, \(m\angle4 + m\angle3 = m\angle4 + m\angle5\), so \(m\angle3 = m\angle5\)