Question

Prove that opposite angles of a cycle quadrilateral are supplementary.​

Answer 1

Given : A cyclic quadrilateral ABCD.
To Prove : ∠A+∠C=180


∠B+∠D=180

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+y =360
....(iv)
From (iii) and (iv), we get
2∠C+2∠A=360

⇒ ∠C+∠A=180

But we know that angle sum property of quadrilateral
∠A+∠B+∠C+∠D=360
∠B+∠D+180=360
∠B+∠D=180

Hence proved.