Cyclic Quadrilaterals

Definition: A cyclic quadrilateral, by definition, is any quadrilateral that can be inscribed inside a circle.  That is, all 4 vertices of a cyclic quadrilateral always lie on the circle itself.   Mess around with the applet for a couple of minutes, and then answer the questions that follow.  
Based upon your observations, what can you conclude about both pairs of opposite angles of any cyclic quadrilateral?   Prove your assertion true using a theorem previously learned.  Explain fully why what you've observed in the applet above is true.