Question

current skill $\\$ find the measure of angle 6.

Explanation:

Step1: Identify Vertical Angles

Angles \(2x + 144\) and \(-7x - 9\) are vertical angles (opposite angles formed by intersecting lines), so they are equal.

$$2x + 144 = -7x - 9$$

Step2: Solve for \(x\)

Add \(7x\) to both sides:

$$2x + 7x + 144 = -9$$

$$9x + 144 = -9$$

Subtract 144 from both sides:

$$9x = -9 - 144$$

$$9x = -153$$

Divide by 9:

$$x = \frac{-153}{9} = -17$$

Step3: Find Angle \(2x + 144\)

Substitute \(x = -17\) into \(2x + 144\):

$$2(-17) + 144 = -34 + 144 = 110$$

Step4: Identify Relationship with Angle 6

Angle 6 and angle \(2x + 144\) are same - side interior angles? Wait, no, looking at the diagram, angle 6 and angle 8 are vertical? Wait, no, angle 6 and angle 2x + 144: Wait, actually, angle 6 and angle (let's see, angle 8 and angle 6: Wait, maybe angle 6 and angle \(2x + 144\) are supplementary? Wait, no, let's re - examine. Wait, angle \(2x + 144\) and angle 8: angle 8 and angle 6 are vertical? Wait, no, let's think about linear pairs. Wait, angle \(2x + 144\) and angle 8: angle 8 and angle 6 are vertical angles? Wait, no, maybe angle 6 and angle \(2x + 144\) are equal? Wait, no, let's correct. Wait, angle \(2x + 144\) and angle 6: Wait, actually, angle \(2x + 144\) and angle 6 are alternate interior angles? Wait, no, the lines are parallel? Wait, the diagram has two pairs of parallel lines? Wait, the two lines with the red arrows are parallel. So angle \(2x + 144\) and angle 6: Wait, no, angle \(2x + 144\) and angle 6: Wait, angle \(2x + 144\) and angle 8 are adjacent (linear pair)? Wait, angle 8 and angle 6 are vertical angles. Wait, angle \(2x + 144\) and angle 8: angle 8 + angle \(2x + 144\)= 180? No, wait, if the lines are parallel, then angle \(2x + 144\) and angle 6 are equal? Wait, no, let's start over.

Wait, we found that \(2x + 144 = 110\) (from step 3). Now, angle 6 and angle (let's see, angle 8: angle 8 and angle 6 are vertical angles? Wait, no, angle 8 and angle 6: angle 8 and angle 6 are vertical? Wait, angle 8 and angle 6: looking at the intersection, angle 8 and angle 6 are vertical angles? Wait, no, angle 8 and angle 6: when two lines intersect, vertical angles are equal. Wait, angle 8 and angle 6: yes, angle 8 and angle 6 are vertical angles. And angle 8 and angle \(2x + 144\): are they supplementary? Wait, no, angle \(2x + 144\) and angle 8: if they are on a straight line, they are supplementary. Wait, angle \(2x + 144\) + angle 8 = 180? But angle 8 and angle 6 are vertical, so angle 8 = angle 6. Wait, maybe I made a mistake. Wait, let's look at the diagram again. The two lines (the non - parallel ones) intersect, creating angles 6, 7, 8, and the angle with \(2x + 144\). Wait, angle \(2x + 144\) and angle 8: angle 8 + angle \(2x + 144\)= 180 (linear pair). Then angle 8 = 180 - 110 = 70. And angle 6 is equal to angle 8 (vertical angles), so angle 6 = 70. Wait, that makes sense.

Wait, let's verify. If \(2x + 144 = 110\), then angle 8 (adjacent to it) is \(180 - 110 = 70\). And angle 6 and angle 8 are vertical angles, so angle 6 = angle 8 = 70.

Answer:

The measure of angle 6 is \(\boldsymbol{70}\) degrees.