Question

given the figure below, find the values of x and z. (15x - 75)° z° (12x - 39)° x = z =

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(15x - 75=12x - 39\).

Step2: Solve for \(x\)

Subtract \(12x\) from both sides: \(15x-12x - 75=12x-12x - 39\), which simplifies to \(3x-75=-39\). Then add 75 to both sides: \(3x-75 + 75=-39 + 75\), so \(3x = 36\). Divide both sides by 3: \(x=\frac{36}{3}=12\).

Step3: Find the measure of one of the vertical angles

Substitute \(x = 12\) into \(15x - 75\): \(15\times12-75=180 - 75=105\).

Step4: Find \(z\)

Since \(z\) and the angle \(15x - 75\) are supplementary (a linear - pair, sum to \(180^{\circ}\)), then \(z=180-(15x - 75)\). Substituting \(x = 12\), we get \(z=180 - 105 = 75\).

Answer:

\(x = 12\)
\(z = 75\)