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. &#x2220BAD + &#x2220BCD = 180⁰
     2. &#x2220ABC + &#x2220ADC = 180⁰

Construction : Join OB and OD
Proof:

1. &#x2220 BOD = 2 ∠BAD || Angle at the centre of a circle is twice the angle at the circumference.
2.  Reflex &#x2220 BOD + &#x2220 BOD = 2 (&#x2220BAD + &#x2220BCD) || Same as stated above.
  or   2 (&#x2220BAD + &#x2220BCD) = 360⁰ || By statement 2.
or   (&#x2220BAD + &#x2220BCD) = 180⁰
3. Similarly by joining OA and OC it can be proved that
  &#x2220ABC + &#x2220ADC = 180⁰
4.   ∴ &#x2220BAD + &#x2220BCD = 180⁰
&      &#x2220ABC + &#x2220ADC = 180⁰   Proved