Question

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°, prove it. Also prove that if one side of a cyclic quadrilateral is produced, then the exterior angle so formed is equal to the interior opposite angles.

Answer 1

Given: A cyclic □ PQRS in which ∠P, ∠R and ∠Q, ∠S are two pairs of opposite angles. 

To prove: ∠P + ∠R = 180° 

∠Q + ∠S = 180° 

Construction: Join O, P and O, R. 

Proof: By Central angle theorem 

∠POR = 2 ∠PSR [For the same segment angle formed at centre of circle is twice the angle formed at remaining part of the circle]

Similarly, ∠P + ∠R = 180° 

Exterior-Angle property of a cyclic Quadrilateral

Given : A cyclic quadrilateral ABCD in which side AB is produced to E.

To prove: ∠CBE = ∠ADC

Proof: ∠ABC + ∠CBE = 180° ... (1)

[By linear pair AXIOM]

Because, by cyclic quadrilateral theorem 

∠ABC + ∠ADC = 180° ... (2)

[Opposite angles of cyclic quadrilateral are supplementary]

From eqn. (1) and (2)

∠ABC + ∠CBE = ∠ABC + ∠ADC

∠CBE = ∠ADC