Question

which equation can be used to solve for x? 3x + 6 = 111 3x + 90 = 180 3x + 96 = 180 3x + 96 = 111

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. The angle \((3x + 6)^{\circ}\) and the non - right angle adjacent to the \(111^{\circ}\) angle are vertical angles.

Step2: Find the measure of the non - right angle adjacent to \(111^{\circ}\)

Since the sum of angles around a point is \(360^{\circ}\), and we have a right - angle (\(90^{\circ}\)) and a \(111^{\circ}\) angle, the non - right angle adjacent to \(111^{\circ}\) and \((3x + 6)^{\circ}\) can be found. But we can also use the fact that the sum of angles on a straight line is \(180^{\circ}\). The angle adjacent to \(111^{\circ}\) on the straight line \(AC\) is \(180 - 111=69^{\circ}\). And since \((3x + 6)^{\circ}\) and this angle are vertical angles, we know that \(3x+6 + 90=180 - 111+90\). In another way, we know that the sum of the angle \((3x + 6)^{\circ}\) and the right - angle and the angle adjacent to \(111^{\circ}\) on the straight line \(AC\) is \(180^{\circ}\). The angle \((3x + 6)^{\circ}\) and the right - angle (\(90^{\circ}\)) together with the other part of the straight - line angle make up \(180^{\circ}\). So \(3x+6 + 90=180\), which simplifies to \(3x+96 = 180\).

Answer:

\(3x + 96 = 180\)