Question

in the figure, the measure of angle 8 is 74°. which angle needs to be equal to 106° for lines n and m to be parallel by the same - side exterior angles converse theorem? options: angle 6, angle 2, angle 1, angle 4

Explanation:

The Same - Side Exterior Angles Converse Theorem states that if two lines are cut by a transversal and the sum of a pair of same - side exterior angles is \(180^{\circ}\), then the two lines are parallel. We know that the measure of angle 8 is \(74^{\circ}\). Let the angle we are looking for be \(x\). Then, according to the theorem, \(x + 74^{\circ}=180^{\circ}\), so \(x = 180^{\circ}- 74^{\circ}=106^{\circ}\).

Now, we need to identify which angle is a same - side exterior angle with angle 8. Angle 1 and angle 8 are same - side exterior angles. Let's verify: Lines \(n\) and \(m\) are cut by the transversal (the horizontal line). Angle 1 is an exterior angle of line \(n\) and angle 8 is an exterior angle of line \(m\), and they are on the same side of the transversal. So if angle 1 is \(106^{\circ}\), then \(106^{\circ}+74^{\circ}=180^{\circ}\), which satisfies the Same - Side Exterior Angles Converse Theorem, making lines \(n\) and \(m\) parallel.

Answer:

angle 1