Question

given the figure below, find the values of x and z. (13x + 31)° (15x + 19)° x = z =

Explanation:

Step1: Set up equation

Since vertical - angles are equal, we set \(13x + 31=15x + 19\).

Step2: Solve for x

Subtract \(13x\) from both sides: \(31 = 2x+19\). Then subtract 19 from both sides: \(2x=31 - 19=12\). Divide both sides by 2: \(x = 6\).

Step3: Find z

First, find one of the angles. Substitute \(x = 6\) into \(13x + 31\), we get \(13\times6+31=78 + 31=109\). Since \(z\) and the angle \((13x + 31)^{\circ}\) are supplementary (a linear - pair, sum to \(180^{\circ}\)), then \(z=180-(13x + 31)\). Substituting \(x = 6\) into the formula for \(z\), we have \(z=180 - 109 = 71\).

Answer:

\(x = 6\)
\(z = 71\)