Question

find the measure of angle 6.

Explanation:

Step1: Identify Parallel Lines and Transversal

The angles \(-3x + 137\) and \(9x - 5\) are corresponding angles (since the lines are parallel, corresponding angles are equal). So we set them equal: \(-3x + 137 = 9x - 5\).

Step2: Solve for \(x\)

Add \(3x\) to both sides: \(137 = 12x - 5\). Then add 5 to both sides: \(142 = 12x\). Divide by 12: \(x=\frac{142}{12}=\frac{71}{6}\)? Wait, no, wait—wait, maybe they are alternate exterior angles? Wait, no, actually, looking at the diagram, the angle \(-3x + 137\) and \(9x - 5\) should be equal because the lines are parallel (assuming the lines with arrows are parallel). Wait, maybe I made a mistake. Wait, let's check again. Wait, the angle \(-3x + 137\) and angle 6—wait, no, angle \(-3x + 137\) and \(9x - 5\): let's solve \(-3x + 137 = 9x - 5\). So \(137 + 5 = 9x + 3x\), \(142 = 12x\), \(x=\frac{142}{12}\)? That can't be. Wait, maybe they are supplementary? Wait, no, if the lines are parallel, corresponding angles are equal. Wait, maybe the angle \(-3x + 137\) and \(9x - 5\) are equal. Wait, let's do the math again: \(-3x + 137 = 9x - 5\). So \(137 + 5 = 9x + 3x\), \(142 = 12x\), \(x = \frac{142}{12} = \frac{71}{6} \approx 11.83\). But that seems odd. Wait, maybe I misidentified the angles. Wait, angle 6 and angle \(-3x + 137\)—wait, angle 6 and angle 8 are vertical angles? No, angle 6 and angle \(-3x + 137\): wait, maybe the angle \(-3x + 137\) and \(9x - 5\) are equal. Wait, let's check the other way. Wait, maybe they are supplementary. Wait, if the lines are parallel, consecutive interior angles are supplementary. Wait, no, let's look at the diagram again. The two angles \(-3x + 137\) and \(9x - 5\) are on the same side of the transversal? No, maybe they are alternate exterior angles. Wait, perhaps the correct equation is \(-3x + 137 + 9x - 5 = 180\) (supplementary). Let's try that. So \(6x + 132 = 180\), \(6x = 48\), \(x = 8\). Ah, that makes sense. So I must have misidentified the angle relationship. So if they are same - side exterior angles, they are supplementary. So \(-3x + 137 + 9x - 5 = 180\). Combine like terms: \(6x + 132 = 180\). Subtract 132: \(6x = 48\), \(x = 8\). Then, angle \(-3x + 137\) when \(x = 8\) is \(-24 + 137 = 113\). Then, angle 6: since angle 6 and angle \(-3x + 137\) are vertical angles? Wait, no, angle 6 and angle 8 are vertical? Wait, angle 6 and \(-3x + 137\): if \(x = 8\), then \(9x - 5 = 72 - 5 = 67\). Then, angle \(-3x + 137 = 113\), and 113 + 67 = 180, so they are supplementary. Then, angle 6: let's see, angle 6 and angle \(-3x + 137\)—wait, angle 6 and angle 8: angle 8 is equal to \(9x - 5\) when \(x = 8\), \(9*8 - 5 = 67\). Wait, angle 6 and angle 8 are adjacent? No, angle 6 and \(-3x + 137\): if the lines intersect, angle 6 and \(-3x + 137\) are vertical angles? Wait, no, the intersection: the two lines intersect, so angle \(-3x + 137\) and angle 6—wait, maybe angle 6 is equal to \(9x - 5\) when \(x = 8\), \(9*8 - 5 = 67\)? Wait, no, let's re - examine.

Wait, the correct approach: the two angles \(-3x + 137\) and \(9x - 5\) are supplementary (since the lines are parallel, consecutive interior angles are supplementary). So:

\(-3x + 137 + 9x - 5 = 180\)

Step1: Combine like terms

\(6x + 132 = 180\)

Step2: Subtract 132 from both sides

\(6x = 180 - 132 = 48\)

Step3: Divide by 6

\(x = 8\)

Now, angle \(-3x + 137\) when \(x = 8\) is \(-3*8 + 137 = -24 + 137 = 113\). Now, angle 6: looking at the diagram, angle 6 and angle \(-3x + 137\)—wait, angle 6 and angle 8: angle 8 is equal to \(9x - 5\) when \(x = 8\), \(9*8 - 5 = 67\). Wait, angle 6 and angle 8 are adjacent? No, angle 6 and \(-3x + 137\): if the two lines intersect, angle 6 and \(-3x + 137\) are vertical angles? No, wait, angle 6 and angle \(-3x + 137\): actually, angle 6 is equal to \(9x - 5\) when \(x = 8\), \(9*8 - 5 = 67\)? Wait, no, let's see: the angle \(-3x + 137\) and angle 6—if the lines are parallel, then angle 6 is equal to \(9x - 5\)? Wait, no, maybe angle 6 is equal to \(9x - 5\). Wait, when \(x = 8\), \(9x - 5 = 67\). Let's check: \(-3x + 137 = 113\), and 113 + 67 = 180, so they are supplementary. Then, angle 6: if angle 6 and \(9x - 5\) are equal? Wait, no, angle 6 and \(-3x + 137\): wait, maybe angle 6 is equal to \(9x - 5\). Wait, let's see the diagram again. The angle labeled \(9x - 5\) and angle 6: if the lines are parallel, then angle 6 is equal to \(9x - 5\). So when \(x = 8\), \(9*8 - 5 = 67\). So angle 6 is 67 degrees.

Wait, let's verify: \(-3x + 137 = -24 + 137 = 113\), \(9x - 5 = 72 - 5 = 67\), and 113 + 67 = 180, which makes sense for supplementary angles (since the lines are parallel, consecutive interior angles are supplementary). Then, angle 6: since angle 6 and \(9x - 5\) are corresponding angles (or alternate interior angles), so angle 6 = \(9x - 5 = 67\).

Answer:

67