Question

find the measure of \\( \angle x \\), \\( \angle y \\), and \\( \angle z \\) in this figure.

Explanation:

Step1: Find \( \angle x \)

Use the exterior - angle property of a triangle. The exterior angle \( x \) is equal to the sum of the two non - adjacent interior angles. In a right - triangle, one non - adjacent interior angle is \( 42^{\circ} \) and the other is \( 90^{\circ} \).
\( x=42^{\circ}+90^{\circ}=132^{\circ} \)

Step2: Find \( \angle y \)

Since \( x \) and \( y \) are supplementary angles (\( x + y=180^{\circ} \))
\( y = 180^{\circ}-x \)
Substitute \( x = 132^{\circ} \) into the equation:
\( y=180^{\circ}-132^{\circ}=48^{\circ} \)

Step3: Find \( \angle z \)

Since \( z \) and \( y \) are vertical angles (vertical angles are equal)
\( z=y = 48^{\circ} \)

Answer:

\( \angle x = 132^{\circ} \), \( \angle y=48^{\circ} \), \( \angle z = 48^{\circ} \)