Question

(i) Prove that opposite angles of a cyclic quadrilateral are supplementary
​

Answer 1

Answer:

Given : A cyclic quadrilateral ABCD.

To Prove : ∠A+∠C=180

o

∠B+∠D=180

o

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 x

o

and y

o

respectively.

Proof : x

o

=2∠C [Angle at centre theorem] ...(i)

y

o

=2∠A ...(ii)

Adding (i) and (ii), we get

x

o

+y

o

=2∠C+2∠A ...(iii)

But, x

o

+y

o

=360

o

....(iv)

From (iii) and (iv), we get

2∠C+2∠A=360

o

⇒ ∠C+∠A=180

o

But we know that angle sum property of quadrilateral

∠A+∠B+∠C+∠D=360

o

∠B+∠D+180

o

=360

o

∠B+∠D=180

o

Hence prove