I am confused on what the reasons are.

Answer:
1) Angle AOC + Angle COB = Angle AOB
Angle AOC = X
Angle COB = Y
putting in equation, we get
X + Y = AOB
2) Angle X + Y = AOB (proved above)
AOB = 90° ( given)
Angle X + Angle y = 90°
Angle X is the complement of Y