Another Obedient Ray

Two mirrors, A and B, are joined at a fixed angle at point O, and a ray of light is shone into the angle between them. It first strikes one mirror at a point X. When the ray re-emerges, the last point it strikes is point Y on one mirror.
  1. If the incident ray is parallel to mirror A, and the exit ray is parallel to mirror B, find the possible angles for AOB and the possible relationships between OX and OY.
  2. Find the requirements on the angle of incidence, with respect to angle AOB, such that OX=OY.

Source: Original. Similar to Ken's POTW January 2, 1997

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