Question

given the figure below, find the values of x and z.
(6x + 36)°
(8x + 22)°
z°
x =
z =

Explanation:

Step1: Identify vertical angles

The angles \((6x + 36)^\circ\) and \((8x + 22)^\circ\) are vertical angles, so they are equal. Thus, we set up the equation:
\(6x + 36 = 8x + 22\)

Step2: Solve for \(x\)

Subtract \(6x\) from both sides:
\(36 = 2x + 22\)
Subtract 22 from both sides:
\(14 = 2x\)
Divide both sides by 2:
\(x = 7\)

Step3: Find the measure of the vertical angles

Substitute \(x = 7\) into \(6x + 36\):
\(6(7) + 36 = 42 + 36 = 78^\circ\)

Step4: Find \(z\)

The angle \(z\) and the vertical angle we found are supplementary (they form a linear pair), so:
\(z + 78 = 180\)
Subtract 78 from both sides:
\(z = 180 - 78 = 102\)

Answer:

\(x = 7\)
\(z = 102\)