Question

what is the value of x? find the measure of angle 3.

Explanation:

Step1: Identify the relationship

The two vertical lines are parallel, and the transversal creates corresponding angles. So \(5x = 60^\circ\) (corresponding angles are equal).

Step2: Solve for \(x\)

Divide both sides by 5: \(x=\frac{60}{5}\)
\(x = 12\)

For the measure of angle 3:

Step1: Identify the relationship

Angle 3 and the \(60^\circ\) angle are same - side interior angles? Wait, no, looking at the diagram, angle 3 and the angle adjacent to \(5x\) (since \(5x = 60^\circ\)): actually, angle 3 and the angle with measure \(5x\) are supplementary? Wait, no, let's re - examine. Wait, the two vertical lines are parallel, and the transversal. Wait, angle 3 and the angle labeled \(60^\circ\): actually, angle 3 and the angle that is \(5x\) (which is \(60^\circ\)): angle 3 and \(5x\) are supplementary? Wait, no, in the diagram, angle 3 and the angle below it (angle 2) are adjacent, and angle 2 is \(5x = 60^\circ\)? Wait, no, maybe I made a mistake. Wait, the two vertical lines are parallel, so the corresponding angles: the angle of \(60^\circ\) and \(5x\) are corresponding angles, so \(5x=60\), so \(x = 12\). Then angle 3: angle 3 and \(5x\) are supplementary? Wait, no, angle 3 and angle 2 (which is \(5x\)): angle 3 + angle 2=180? Wait, no, if the two vertical lines are parallel, and the transversal, then angle 3 and the angle with \(60^\circ\): wait, maybe angle 3 is equal to \(180 - 60=120^\circ\)? Wait, no, let's start over.

Wait, the two vertical lines are parallel. The transversal cuts them. The angle of \(60^\circ\) and \(5x\) are corresponding angles, so \(5x = 60\), so \(x = 12\). Then angle 3: angle 3 and \(5x\) are supplementary (linear pair), so angle 3 \(=180 - 5x\). Substitute \(x = 12\), \(5x=60\), so angle 3 \(=180 - 60 = 120^\circ\).

Answer:

For \(x\): \(x = 12\)
For the measure of angle 3: \(120^\circ\)