Question

consider the figure, where line segment mn is parallel to line segment pq. which statement must be true? o since m∠1 + m∠2 + m∠3 = 180, m∠1 = m∠4, and m∠3 = m∠5, we can prove that m∠4 + m∠2 + m∠5 = 180. o since m∠1 + m∠2 + m∠3 = 180, m∠1 = m∠5, and m∠3 = m∠4, we can prove that m∠5 + m∠2 + m∠4 = 180. o since m∠1 + m∠2 + m∠3 = 360, m∠1 = m∠4, and m∠3 = m∠5, we can prove that m∠5 + m∠2 + m∠4 = 180. o since m∠1 + m∠2 + m∠3 = 360, m∠1 = m∠5, and m∠3 = m∠4, we can prove that m∠4 + m∠2 + m∠5 = 360.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180 degrees. In \(\triangle MNP\), \(m\angle1 + m\angle2 + m\angle3=180\).

Step2: Use the property of parallel lines

Since \(MN\parallel PQ\), we have alternate - interior angles equal. That is, \(\angle1\) and \(\angle4\) are alternate - interior angles, so \(m\angle1 = m\angle4\), and \(\angle3\) and \(\angle5\) are alternate - interior angles, so \(m\angle3 = m\angle5\).

Step3: Substitute the equal angles

Substitute \(m\angle1\) with \(m\angle4\) and \(m\angle3\) with \(m\angle5\) in the equation \(m\angle1 + m\angle2 + m\angle3 = 180\). We get \(m\angle4 + m\angle2 + m\angle5=180\).

Answer:

Since \(m\angle1 + m\angle2 + m\angle3 = 180\), \(m\angle1 = m\angle4\), and \(m\angle3 = m\angle5\), we can prove that \(m\angle4 + m\angle2 + m\angle5 = 180\).