Question

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

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So, \(9x + 57=5x + 89\).

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(9x-5x + 57=5x-5x + 89\), which gives \(4x+57 = 89\).
Subtract \(57\) from both sides: \(4x+57 - 57=89 - 57\), so \(4x=32\).
Divide both sides by \(4\): \(x=\frac{32}{4}=8\).

Step3: Find \(z\)

Use the linear - pair property (adjacent angles on a straight line sum to \(180^{\circ}\)).
Take one of the angles, say \(9x + 57\). Substitute \(x = 8\): \(9\times8+57=72 + 57=129\).
Then \(z=180-(9x + 57)\) (or \(z = 180-(5x + 89)\)). Substitute \(x = 8\) into \(z=180-(9x + 57)\): \(z=180 - 129=51\).

Answer:

\(x = 8\), \(z = 51\)