HMMT 1999 O1

Start with an angle of $60^{\circ}$ and bisect it, then bisect the lower $30^{\circ}$ angle, then the upper $15^{\circ}$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^{\circ}$ angle into two angles. Find the measure (degrees) of the smaller angle.