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

To prove:

∠ABC + ∠ADC = 180°

and ∠BAD + ∠BCD = 180°

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 = 1\over 2 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 = 1\over 2 ∠AOC ……….(ii)

From (i) and (ii), we get:

∠ABC + ∠ADC = 1\over 2 reflex ∠AOC + 1\over 2 ∠AOC

= 1\over 2 (reflex ∠AOC + ∠AOC)

= 1\over 2 × 4 right angles

= 180°

Similarly, by joining B, O, and D, O; it can be proved that,

∠BAD + ∠BCD = 180° (proved).