Question

find the measures of angles x, y, and z in the figure.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. Angle $x$ and the $105^{\circ}$ angle are vertical angles, so $x = 105^{\circ}$.

Step2: Use linear - pair property

The sum of angles in a linear pair is $180^{\circ}$. Angle $y$ and the $105^{\circ}$ angle form a linear pair. So $y=180 - 105=75^{\circ}$.

Step3: Use vertical - angle property again

Angle $z$ and angle $y$ are vertical angles. So $z = y=75^{\circ}$.

Answer:

$x = 105^{\circ}$, $y = 75^{\circ}$, $z = 75^{\circ}$