Question

given the figure below, find the values of x and z.
(11x - 41)°
z°
(14x - 71)°
x =
z =

Explanation:

Step1: Identify Vertical Angles

The two angles \((11x - 41)^\circ\) and \((14x - 71)^\circ\) are vertical angles, so they are equal. Thus, we set up the equation:
\(11x - 41 = 14x - 71\)

Step2: Solve for \(x\)

Subtract \(11x\) from both sides:
\(-41 = 3x - 71\)
Add \(71\) to both sides:
\(30 = 3x\)
Divide by \(3\):
\(x = 10\)

Step3: Find the Angle Measure

Substitute \(x = 10\) into one of the angle expressions, say \(11x - 41\):
\(11(10) - 41 = 110 - 41 = 69^\circ\)

Step4: Find \(z\)

The angle \(z^\circ\) and the \(69^\circ\) angle are supplementary (they form a linear pair), so:
\(z + 69 = 180\)
Subtract \(69\) from both sides:
\(z = 111\)

Answer:

\(x = 10\), \(z = 111\)