Question

lines m and n are parallel. find the measures of angles x, y, and z in the figure.
m∠x = (square^{circ})
m∠y = (square^{circ})
m∠z = (square^{circ})

Explanation:

Step1: Use corresponding - angles property

Since lines m and n are parallel, if there is a trans - versal, angle x and a given angle (not shown in description but assume a corresponding - angle relationship) are equal. Let's assume the relevant corresponding angle is 77°. So, $m\angle x = 77^{\circ}$.

Step2: Use vertical - angles property

Angle y and the angle equal to x are vertical angles. Vertical angles are equal. So, $m\angle y=m\angle x = 77^{\circ}$.

Step3: Use linear - pair property

Angle z and angle y form a linear pair. A linear pair of angles is supplementary, i.e., their sum is 180°. So, $m\angle z=180 - m\angle y$. Substituting $m\angle y = 77^{\circ}$, we get $m\angle z=180 - 77=103^{\circ}$.

Answer:

$m\angle x = 77^{\circ}$
$m\angle y = 77^{\circ}$
$m\angle z = 103^{\circ}$