Question

Prove that “The opposite angles of a cyclic quadrilateral are supplementary”.

Answer 1

ABCD is a cyclic quadrilateral of the circle with centre O. 

We know that the angle subtended by the arc at the centre is double the angle subtended by it at the remaining part of the circle.

Thus, 

\[\angle\] DOB = 2 \[\angle\]DAB                     .....(1)

Also, reflex\[\angle\]DOB = 2 \[\angle\]DCB            .....(2) 

Adding (1) and (2), we get

2\[\angle\]DAB + 2\[\angle\]DCB = \[\angle\] DOB + reflex\[\angle\]DOB

\[\Rightarrow\]2(\[\angle\]DAB + \[\angle\]DCB) = `360^o`

\[\Rightarrow \angle DAB + \angle DCB = \frac{360^o}{2}\]

\[ \Rightarrow \angle DAB + \angle DCB = 180^o\]

Hence proved.