The opposite angles of quadrilaterals in circles are equal to two right angles.
Proof
Let ABCD be a cyclic quadrilateral. Join AC and BD. By
III.21, ∠BAC=∠BDC (both subtend arc BC from the
opposite side), and ∠CAD=∠CBD (both subtend arc
CD). So ∠BAD=∠BAC+∠CAD=∠BDC+∠CBD.
In △BCD, by I.32 the three angles sum to two right
angles: ∠BDC+∠CBD+∠BCD= two right angles.
Substituting ∠BDC+∠CBD=∠BAD:
∠BAD+∠BCD= two right angles.