Question

In the question Prove that opposite angle of a cyclic quadrilateral are supplimentary ​

Answer 1

Answer:

Given :- A cyclic quadrilateral ABCD.

To prove :- ∠A+∠C = 180⁰

                 ∠B+∠D = 180⁰

Construction :- Let O be the centre of the circle. Join O to B and D. Then let the angle subtended by the minor arc and the major arc at the centre be xo and yo respectively.

Proof :- xo=2∠C      [Angle at centre theorem]    ...(i)

           yo=2∠A                                                   ...(ii)

Adding (i) and (ii), we get

           xo+yo=2∠C+2∠A                  ...(iii)

But,     xo+yo=360⁰                 ....(iv)

From (iii) and (iv), we get

            2∠C +2∠A= 360⁰

⇒            ∠C +∠A= 180⁰

But we know that angle sum property of quadrilateral

∠A +∠B +∠C +∠D = 360⁰

∠B + ∠D + 180⁰ = 360⁰

∠B+∠D=180⁰

Hence proved.