Hi Mel,

Angle x is 1/2 the sum of its intercepted arcs or 1/2 (30 + 134).

Arc BE measures 118 because it is twice the measure of the inscribed angle D which is 59 degrees. Therefore, angle y is 1/2 the measure of arc BE because it is also an inscribed angle. Angle y is the same measure as angle D because they both intercept the same arc BE.

To find the measure of arc DF we have to subtract the other arcs from 360. So 360 - 134 - 118 - 30 = 78 degrees for arc DF. And angle z is 1/2 of that.