Question

given the figure below, find the values of x and z.
(7x + 65)° (12x + 30)°
z°
x =
z =

Explanation:

Step1: Identify Vertical Angles or Supplementary Angles

The two angles \((7x + 65)^\circ\) and \((12x + 30)^\circ\) are supplementary? Wait, no, when two lines intersect, adjacent angles are supplementary, but vertical angles are equal. Wait, looking at the diagram, the two angles \((7x + 65)^\circ\) and \((12x + 30)^\circ\) are actually supplementary? Wait, no, if they are adjacent and form a linear pair, then they should be supplementary. Wait, no, let's check: when two lines intersect, the sum of adjacent angles is \(180^\circ\). Wait, but maybe they are vertical angles? Wait, no, the labels: let's see, the angle \((7x + 65)\) and \((12x + 30)\) – maybe they are supplementary? Wait, no, let's solve for \(x\) first. Wait, maybe they are equal? Wait, no, if they are vertical angles, but maybe I made a mistake. Wait, let's assume that the two angles \((7x + 65)^\circ\) and \((12x + 30)^\circ\) are supplementary? Wait, no, let's check:

Wait, actually, when two lines intersect, the adjacent angles are supplementary. Wait, but maybe the two angles \((7x + 65)\) and \((12x + 30)\) are supplementary? Wait, no, let's set them equal? Wait, no, maybe I messed up. Wait, let's think again. The two angles given are \((7x + 65)^\circ\) and \((12x + 30)^\circ\). If they are vertical angles, they should be equal. Wait, but maybe they are supplementary? Wait, no, vertical angles are equal. Wait, let's try setting them equal:

\(7x + 65 = 12x + 30\)

Wait, solving that:

\(65 - 30 = 12x - 7x\)

\(35 = 5x\)

\(x = 7\)

Wait, but then let's check the angles:

\(7(7) + 65 = 49 + 65 = 114^\circ\)

\(12(7) + 30 = 84 + 30 = 114^\circ\). Oh! So they are equal, so they are vertical angles. So that means \(7x + 65 = 12x + 30\)

Step2: Solve for \(x\)

\(7x + 65 = 12x + 30\)

Subtract \(7x\) from both sides:

\(65 = 5x + 30\)

Subtract 30 from both sides:

\(35 = 5x\)

Divide both sides by 5:

\(x = 7\)

Step3: Find \(z\) (Wait, the problem says \(z\), but in the diagram, maybe it's a typo, or maybe \(z\) is the other angle. Wait, after finding \(x\), we can find the measure of the angle, and then \(z\) would be equal to one of the vertical angles or supplementary. Wait, let's check:

After \(x = 7\), the angle \((7x + 65) = 7*7 + 65 = 49 + 65 = 114^\circ\). Then the adjacent angle \(z\) would be supplementary to 114°, so \(z = 180 - 114 = 66^\circ\)? Wait, no, maybe \(z\) is the vertical angle? Wait, the diagram has \(z^\circ\) as the angle adjacent? Wait, maybe I misread. Wait, the problem says "find the values of \(x\) and \(z\)". Let's re-examine.

Wait, when two lines intersect, the vertical angles are equal, and adjacent angles are supplementary. So if \((7x + 65)^\circ\) and \((12x + 30)^\circ\) are vertical angles (equal), then after finding \(x = 7\), the angle is 114°, then \(z\) is the adjacent angle, so \(z = 180 - 114 = 66^\circ\)? Wait, but maybe \(z\) is the vertical angle to one of them? Wait, no, the diagram shows \(z^\circ\) as the angle opposite to... Wait, maybe the problem has a typo, and \(z\) is actually the other angle. Wait, let's proceed.

First, solve for \(x\):

From \(7x + 65 = 12x + 30\)

\(65 - 30 = 12x - 7x\)

\(35 = 5x\)

\(x = 7\)

Then, the angle \((7x + 65) = 114^\circ\), so the adjacent angle \(z\) (if it's a linear pair) would be \(180 - 114 = 66^\circ\). So \(z = 66\).

Answer:

\(x = 7\), \(z = 66\)