There are two parallel lines and a transversal cutting through both lines. Looking at the diagram,
angle 64 and angle x lie in similar positions on the parallel lines but on alternate sides of the transversal. This means that they are alternate angles. Alternate angles are equal. Hence,
angle x = 64 degrees
angle z and angle y lie on a straight line. they are linear pairs. The sum of the angles on a straight line is 180 degrees. This means that
x + y = 180
64 + y = 180
y = 180 - 64
angle y = 116 degrees
Angle x and angle z are vertically opposite angles. vertically opposite angles are equal. This means that
angle z = 64 degrees