Question

1. determine the measures of the unknown angles in the figure.

Explanation:

Step1: Find angle \(w\) using triangle - angle sum

In the left - hand triangle, the sum of interior angles of a triangle is \(180^{\circ}\). So \(w=180-(42 + 85)\).
\[w=180 - 127=53^{\circ}\]

Step2: Use vertical angles property

Since \(x\) and \(w\) are vertical angles, vertical angles are equal. So \(x = w\).
\[x = 53^{\circ}\]

Step3: Find angle \(y\) using triangle - angle sum

In the right - hand triangle, the sum of interior angles of a triangle is \(180^{\circ}\). So \(y=180-(88 + 53)\).
\[y=180 - 141 = 39^{\circ}\]

Answer:

\(w = 53^{\circ}\), \(x = 53^{\circ}\), \(y = 39^{\circ}\)