Theorem : Opposite Angles of a Cyclic Quadrilateral are Supplementary | Class 8

Theorem : Opposite Angles of a Cyclic Quadrilateral are Supplementary.

Solution :

Given: ABCD is a cyclic quadrilateral.
R.T.P.(Require To Prove)

1. ∠BAD + ∠BCD = 180⁰
     2. ∠ABC + ∠ADC = 180⁰

Construction : Join OB and OD
Proof:

1. ∠ BOD = 2 ∠BAD || Angle at the centre of a circle is twice the angle at the circumference.
2.  Reflex ∠ BOD + ∠ BOD = 2 (∠BAD + ∠BCD) || Same as stated above.
  or   2 (∠BAD + ∠BCD) = 360⁰ || By statement 2.
or   (∠BAD + ∠BCD) = 180⁰
3. Similarly by joining OA and OC it can be proved that
  ∠ABC + ∠ADC = 180⁰
4.   ∴ ∠BAD + ∠BCD = 180⁰
&      ∠ABC + ∠ADC = 180⁰   Proved