Question

The sum of either pair of opposite angles of a cyclic quadrilateral is 180⁰ . Prove that.

Answer 1

Given: A cyclic quadrilateral ABCD. 

To Prove : ∠A + ∠C = ∠B + ∠D = 180⁰ 

Construction : Join AC and BD. 

Proof : ∠ACB = ∠ADB [Angles of same segment] 

And ∠BAC = ∠BDC [Angles of same segment] 

∴ ∠ACB + ∠BAC = ∠ADB + ∠BDC =∠ADC. 

 Adding ∠ABC to both sides, we get 

∠ACB + ∠BAC + ∠ABC =∠ADC + ∠ABC. 

The left side being the sum of three angles of ∆ABC is equal to 180⁰ . 

∴ ∠ADC + ∠ABC = 180⁰ 

 i.e., ∠D + ∠B = 180⁰ 

∴ ∠A + ∠C = 360⁰ -(∠B + ∠D) = 180⁰ [∴ ∠A + ∠B + ∠C + ∠D = 360⁰ ]