Question

Prove that opposite angles of a cyclic quadrilateral are supplementary

Answer 1

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle. An example is pictured below: Prove that the opposite angles in a cyclic quadrilateral that contains the center of the circle are supplementary.

Answer 2

Hence, it is proved that the opposite angles of a cyclic quadrilateral are supplementary(sum of 180°).

Step-by-step explanation:

Let PQRS be the cyclic quadrilateral

To prove:

∠P + ∠R = 180°

∠Q + ∠S = 180°

Construction: Assume O as the centre of the circle. Now, Join O to Q and S. Then allow the angle subtended by the major and minor arc be a° and b°

Proof:

a° = 2∠R          ...(1)

b° = 2∠P          ...(2)                              (∵ Angle at center theorem)

By adding the equations (1) & (2),

a° and b° = 2∠R + 2∠P    ...(3)

yet a° + b° = 360°   ...(4)

From equations (3) & (4), we get

2∠R + 2∠P = 360°

∠R + ∠P = 360°/2

∵ ∠R + ∠P = 180°

Now,

∠P + ∠Q + ∠R + ∠S = 360°     (∵ angle sum property of a quadrilateral)

∠Q + ∠S + 180° = 360°                    (∵ ∠R + ∠P = 180°)

∠Q + ∠S = 360° - 180°

∵ ∠Q + ∠S = 180°

Hence proved that opposite angles of a cyclic quadrilateral are supplementary.

Learn more: Supplementary angles

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