Since the angle at the center of the circle is twice the angle at the circumference that subtends it, we can find the measure of each arc by finding the central angle that subtends each arc and then doubling that angle.
The central angle that subtends arc AB is 60 degrees, so arc AB measures 2 × 60 = 120 degrees.
The central angle that subtends arc AC is 90 degrees, so arc AC measures 2 × 90 = 180 degrees.
The central angle that subtends arc BC is 210 degrees, so arc BC measures 2 × 210 = 420 degrees.
Therefore, arc AB measures 120 degrees, arc AC measures 180 degrees, and arc BC measures 420 degrees.