A cyclic quadrilateral ABCD has parallel chords AB and DC. If angle C is 50 degrees, find angles x and z.

The figure shows a cyclic quadrilateral (all vertices lie on the circumference of the circle). Let the vertices be A (top-left, angle $x$), B (top-right, angle $y$), C (bottom-right, angle $50^\circ$), and D (bottom-left, angle $z$).
The arrows on chords AB and DC indicate that these chords are parallel ($\text{AB} \parallel \text{DC}$).

If two chords in a circle are parallel, then the arcs intercepted between them are equal. Therefore, arc AD = arc BC.