Question

Prove that in a cyclic quadrilateral, the exterior angle at any vertex is equal to the interior angle at the opposite vertex.

Answer 1

Consider the cyclic quadrilateral ABCD whose one side DC is extended to point E.

We know that opposite angles of a cyclic quadrilateral are supplementary.

∴ ∠DAB + ∠DCB = 180°

⇒ ∠DCB = 180° − ∠DAB … (1)

We know that sum of the angles forming a linear pair is 180°.

∴ ∠BCE + ∠BCD = 180°

⇒ ∠BCD = 180° − ∠BCE … (2)

From equation (1) and equation (2), we get:

180° − ∠DAB = 180° − ∠BCE

⇒ −∠DAB = −∠BCE

⇒ ∠DAB = ∠BCE

Hence, in a cyclic quadrilateral, the exterior angle at any vertex is equal to the interior angle at the opposite vertex.