Given: ABCD is a cyclic quadrilateral of a circle with centre O.

To prove: ∠ABC + ∠ADC = 2 right angles and ∠BAD + ∠BCD = 2 right angles

Construction: A, O and C, O are joined.

Proof: The reflex angle ∠AOC at the centre and the angle ∠ABC on the circle are formed with the circular arc ADC.

∴ Reflex ∠AOC = 2∠ABC

∴ ∠ABC = ½ reflex ∠AOC ………(i)

Again, ∠AOC is the angle at the centre and ∠ADC is the angle on the circle formed with the circular arc ABC.

∴ ∠AOC = 2∠ADC

∴ ∠ADC = ½ ∠AOC ………(ii)

∴ From (i) and (ii), we get

∠ABC + ∠ADC = ½ reflex ∠AOC + ½ ∠AOC

= ½ (reflex ∠AOC + ∠AOC)

= ½ × 4 right angles = 2 right angles

Similarly, by joining B, O and D, O; it can be proved that, ∠BAD + ∠BCD = 2 right angles (Proved)