Find that Angle! (GoGeometry Action 144)

Creation of this applet was inspired by a tweet posted by Antonio Gutierrez (GoGeometry). Geometry Students: Take a look at the applet below. What is the measure of the pink angle? This problem may look difficult at first, but it's actually quite simple. As a hint, try drawing this diagram on a separate sheet of paper. Yet on this paper, draw the ENTIRE CIRCLE (of which this 1/4 circle is a part). Any insight? See if you can figure this out WITHOUT sliding the "Solution" slider.