Question

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

Explanation:

Step1: Set up the equation

Vertical - angles are equal. So, we set \(7x + 51=10x+33\).

Step2: Rearrange the equation

Subtract \(7x\) from both sides: \(51 = 10x - 7x+33\), which simplifies to \(51 = 3x+33\).

Step3: Solve for \(x\)

Subtract 33 from both sides: \(51 - 33=3x\), so \(18 = 3x\). Then divide both sides by 3, we get \(x = 6\).

Step4: Find the measure of the angles

Substitute \(x = 6\) into \(7x + 51\): \(7\times6+51=42 + 51=93^{\circ}\).

Step5: Find \(z\)

Since the sum of angles around a point is \(360^{\circ}\) and the two vertical - angle pairs are equal, and we know one pair of vertical angles is \(93^{\circ}\) each. Let the other pair of vertical angles be \(z\) each. So, \(2\times93+2z = 360\). First, calculate \(2\times93 = 186\). Then the equation becomes \(186+2z = 360\). Subtract 186 from both sides: \(2z=360 - 186 = 174\). Divide both sides by 2, we get \(z = 87\).

Answer:

\(x = 6\), \(z = 87\)