Question

find the measure of the missing angles.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle opposite the \(114^{\circ}\) angle has the same measure. Let's first find angle \(e\). Since vertical angles are equal, \(e = 114^{\circ}\).

Step2: Use angle - addition property

The sum of angles around a point is \(360^{\circ}\). We know one angle is \(114^{\circ}\) and another is \(22^{\circ}\), and we want to find \(d\) and \(f\). First, consider the right - angle formed by the vertical lines. Let's assume the sum of the non - right - angle parts of the angles around the point is \(x\). We know that \(x+ 90^{\circ}+90^{\circ}=360^{\circ}\), so \(x = 180^{\circ}\).
We know that \(114^{\circ}+22^{\circ}+d = 180^{\circ}\). Then \(d=180^{\circ}-(114^{\circ} + 22^{\circ})=44^{\circ}\).
Since \(d\) and \(f\) are vertical angles, \(f = d = 44^{\circ}\).

Answer:

\(d = 44^{\circ}\), \(e = 114^{\circ}\), \(f = 44^{\circ}\)