Question

given the figure below, find the values of x and z.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(5x + 9=4x - 3\).

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(5x-4x + 9=4x-4x - 3\), which gives \(x+9=-3\). Then subtract 9 from both sides: \(x=-3 - 9=-12\).

Step3: Find \(z\)

Since the sum of angles around a point is \(360^{\circ}\) and the vertical - angle pairs are equal, and we know that the two given angles are vertical angles. Also, \(z\) is supplementary to either of the given angles. Let's use one of the angles, say \(5x + 9\). Substitute \(x = - 12\) into \(5x+9\): \(5\times(-12)+9=-60 + 9=-51\). Then \(z = 180-(-51)=180 + 51 = 231\) (but angles are usually considered between \(0^{\circ}\) and \(360^{\circ}\), and since the non - reflex angle adjacent to \(5x + 9\) is what we want, \(z=180-(5x + 9)\) or \(z=180-(4x - 3)\)). Substitute \(x=-12\) into \(z = 180-(5x + 9)\): \(z=180-(5\times(-12)+9)=180-(-60 + 9)=180 + 51=129^{\circ}\).

Answer:

\(x=-12\), \(z = 129^{\circ}\)