Question

Prove that the oppositesides of a quadliteral circumscrbing a circle

Answer 1

Let ABCD be a quadrilateral circumscribing a circle with centre O.  Now join AO, BO, CO, DO.From the figure, ∠DAO = ∠BAO [Since, AB and AD are tangents]  Let ∠DAO = ∠BAO = 1  Also ∠ABO = ∠CBO [Since, BA and BC are tangents]  Let ∠ABO = ∠CBO = 2 Similarly we take the same way for vertices C and D Sum of the angles at the centre is 360°  Recall that sum of the angles in quad. ABCD = 360°  ⇒ 2(1 + 2 + 3 + 4) = 360°  ⇒ 1 + 2 + 3 + 4 = 180°  In ΔAOB, ∠BOA = 180 – (a + b) In ΔCOD, ∠COD = 180 – (c + d) Angle BOA + angle COD = 360 – (a + b + c + d)  = 360°  – 180°  = 180°  Hence AB and CD subtend supplementary angles at O Thus, opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.