Question

1. find the unknown angle measure (x, y, and z) as per the figure given below.

67° 44°
35°
44°
66°
26°
z°

Explanation:

Step1: Find the measure of ∠4

Use the triangle angle - sum theorem (\(180^{\circ}\) in a triangle).
For the top - triangle: \(\angle2+\angle3+\angle4 = 180^{\circ}\). Given \(\angle2 = 35^{\circ}\), \(\angle3 = 74^{\circ}\).
So, \(\angle4=180^{\circ}-(35^{\circ}+74^{\circ})=180^{\circ}-109^{\circ} = 71^{\circ}\).

Step2: Use the vertical - angles property

\(\angle4\) and the non - labeled angle in the bottom - triangle are vertical angles. So they are equal. Let the non - labeled angle in the bottom - triangle be \(x\), then \(x = \angle4=71^{\circ}\).

Step3: Find the measure of \(z\)

For the bottom - triangle, use the triangle angle - sum theorem (\(180^{\circ}\) in a triangle).
\(z + 26^{\circ}+66^{\circ}=180^{\circ}\).
\(z=180^{\circ}-(26^{\circ}+66^{\circ})=180^{\circ}-92^{\circ}=88^{\circ}\).

Answer:

\(88^{\circ}\)