Draw a line from Y through the center of the circle to the opposite edge. Label the opposite edge T. ZYT = 90o. That's a property of a tangent line and a diameter touching the tangent point.
ZYX = 60o Given
TYX + ZYX = 90o Property of a diameter and circles. That's
TYX = 90 - 60
TYX = 30.
The arc measurement between T and X = 60o Property of a central angle and an angle whose vertex is on the circumference sharing the same endpoints of chord are related by a ratio of 2:1.
The rest of the arc = 120o. The rest of the arc is 180 - 60 = 120. The rest of the arc is XYZ which is 120o
Arc XYZ is bisected by YW. Therefore Arc XW = 120/2 = 60
