Question

given the figure below, find the values of x and z. (11x + 35)° (5x + 81)° z° x = z =

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. So, \(11x + 35=5x + 81\).

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(11x-5x + 35=5x-5x + 81\), which simplifies to \(6x+35 = 81\). Then subtract 35 from both sides: \(6x+35 - 35=81 - 35\), giving \(6x=46\). Divide both sides by 6: \(x=\frac{46}{6}=\frac{23}{3}\).

Step3: Find \(z\)

The sum of angles around a point is \(360^{\circ}\), and the two pairs of vertical angles sum to \(360^{\circ}\). Also, \(z\) is supplementary to either \(11x + 35\) or \(5x + 81\). First, find the measure of \(11x+35\) with \(x = \frac{23}{3}\): \(11\times\frac{23}{3}+35=\frac{253}{3}+35=\frac{253 + 105}{3}=\frac{358}{3}\). Then \(z = 180-(5x + 81)\). Substitute \(x=\frac{23}{3}\) into \(5x + 81\): \(5\times\frac{23}{3}+81=\frac{115}{3}+81=\frac{115+243}{3}=\frac{358}{3}\). So \(z=180-\frac{358}{3}=\frac{540 - 358}{3}=\frac{182}{3}\).

Answer:

\(x=\frac{23}{3}\), \(z=\frac{182}{3}\)