QQuestion
Prove that the opposite angles of a cyclic quadrilateral are supplementary.
✓Answer
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)
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