Question

use the diagram to the right to write a flow proof for the following. given that m∠10 + m∠5 = 180°, m∠10 + m∠3 = 180°, prove that d||s and c||h.

Explanation:

Step1: Recall angle - sum properties

We know that if the sum of two adjacent angles is 180°, they are supplementary. Given \(m\angle10 + m\angle5=180^{\circ}\), \(\angle10\) and \(\angle5\) are supplementary.

Step2: Apply parallel - line theorem

If two lines are cut by a transversal and a pair of same - side interior angles are supplementary, then the two lines are parallel. So, \(q\parallel s\) (since \(\angle10\) and \(\angle5\) are same - side interior angles).

Step3: Use the other given angle sum

Given \(m\angle10 + m\angle3 = 180^{\circ}\), \(\angle10\) and \(\angle3\) are supplementary.

Step4: Apply parallel - line theorem again

Since \(\angle10\) and \(\angle3\) are same - side interior angles and they are supplementary, \(c\parallel h\).

Answer:

We have proven that \(q\parallel s\) and \(c\parallel h\) using the fact that if same - side interior angles are supplementary, then the lines are parallel.