Question

prove that sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree

Answer 1

Given: Cyclic quadrilateral ABCD
To Prove: angle A + angle C =180 degree.
Firstly we join the AC and BD.
Proof: Consider side AB of quadrilateral ABCD as the chord of the circle.

angle ACB= angle ADB ..... 1
angle BAC= angle BDC ..... 2
Adding 1 and 2,we get

angle ACB + angle BAC = angle ADB + angle BDC
angle ACB + angle BAC = angle ADC

angle ABC + angle ACB + angle BAC = angle ABC + angle ADC
180 degree = angle ABC + angle ADC

angle ABC + angle ADC = 180 degree
angle But + angle D = 180 degree

But angle A + angle B + angle C + angle D=
360 degree


Hence, angle A + angle C = 360 -( angle B +
angle D)

angle A + angle C = 360 -180= 180

Hence, angle A + angle C = 180 degree

and angle B + angle D = 180 degree.


Q. E. D.

Answer 2

Given: Cyclic quadrilateral ABCD

To Prove: angle A + angle C =180 degree.

Firstly we join the AC and BD.

Proof: Consider side AB of quadrilateral ABCD as the chord of the circle.

angle ACB= angle ADB ..... 1

angle BAC= angle BDC ..... 2

Adding 1 and 2,we get

angle ACB + angle BAC = angle ADB + angle BDC

angle ACB + angle BAC = angle ADC

angle ABC + angle ACB + angle BAC = angle ABC + angle ADC

180 degree = angle ABC + angle ADC

angle ABC + angle ADC = 180 degree

angle But + angle D = 180 degree

But angle A + angle B + angle C + angle D=

360 degree

Hence, angle A + angle C = 360 -( angle B +

angle D)

angle A + angle C = 360 -180= 180

Hence, angle A + angle C = 180 degree

and angle B + angle D = 180 degree.

Q. E. D.