Question

use the diagram to the right to write a flow proof for the following. given that m∠8 + m∠9 = 180°, m∠8 + m∠3 = 180°, prove that n∥b and g∥s.
statement reason
m∠8 + m∠9 = 180° given
m∠8 + m∠3 = 180° given

Explanation:

Step1: Set up the equations

We are given \(m\angle8 + m\angle9=180^{\circ}\) and \(m\angle8 + m\angle3 = 180^{\circ}\).

Step2: Use the substitution property

Since \(m\angle8 + m\angle9=180^{\circ}\) and \(m\angle8 + m\angle3 = 180^{\circ}\), we can say \(m\angle8 + m\angle9=m\angle8 + m\angle3\). Subtracting \(m\angle8\) from both sides gives \(m\angle9=m\angle3\). These are corresponding - angles. If corresponding angles are equal, then \(n\parallel b\) (Corresponding Angles Postulate).

Step3: Analyze other angle - relationships

We also know that \(m\angle8 + m\angle9 = 180^{\circ}\) and \(m\angle8 + m\angle3 = 180^{\circ}\). \(\angle8\) and \(\angle9\) are same - side interior angles, and \(\angle8\) and \(\angle3\) are same - side interior angles. Since \(m\angle8 + m\angle9 = 180^{\circ}\) and \(m\angle8 + m\angle3 = 180^{\circ}\), for lines \(g\) and \(s\), the same - side interior angles are supplementary. So \(g\parallel s\) (Same - Side Interior Angles Postulate).

Answer:

We have proved that \(n\parallel b\) and \(g\parallel s\) using the given angle - sum equalities and angle - relationship postulates (Corresponding Angles Postulate and Same - Side Interior Angles Postulate).