Question

find the values of the missing angles in the diagram below.

Explanation:

Step1: Find the value of \(w\)

\(w\) and the \(40^{\circ}\) angle are supplementary (a straight - line angle is \(180^{\circ}\)). So \(w = 180 - 40=140^{\circ}\).

Step2: Find the value of \(z\)

\(z\) and \(w\) are vertical angles. Vertical angles are equal. So \(z = w = 140^{\circ}\).

Step3: Find the value of \(x\)

The angle adjacent to the \(115^{\circ}\) angle is \(180 - 115 = 65^{\circ}\) (supplementary angles). Consider the triangle with angles \(x\), \(65^{\circ}\), and the angle adjacent to the \(93^{\circ}\) angle (which is \(180 - 93=87^{\circ}\)). Using the angle - sum property of a triangle (\(180^{\circ}\) in a triangle), \(x=180-(65 + 87)=28^{\circ}\).

Step4: Find the value of \(y\)

\(y\) and \(x\) are vertical angles. Vertical angles are equal. So \(y = x = 28^{\circ}\).

Answer:

\(w = 140\), \(x = 28\), \(y = 28\), \(z = 140\)