Question

Show that if the opposite angles of a quadrilateral add up to two right angles, then the vertices are concyclic

Answer 1

Ans:

Given : We have a quadrilateral let it be ABCD

Also the sum of the opposite angles of this quadrilateral is as shown below

Here

Now as we know that we can draw a circle passing throw any three non collinear points hence we will draw a circle passing throw the points A, B and C as shown below

Here we assume that the point D is not on the circle

Now we will draw a cyclic quadrilateral ABCO such that th epoint O lies on the circle and the side AD of the given quadrilateral as shown below

Now according to the diagram ABCO is a cyclic quadrilateral

As these are the opposite angles of a cyclic quadrilateral hence they are supplementary to each other

But in the question it is given that

Now in triangle BOD

(Property of an exterior angle)

Therefore the points D and O coinsides that means both the points are the same

Therefore our assumption is wrong

Therefore the point D lies on the circle

and hence the given quadrilateral ABCD is a cyclic quadrilateral or its vertices are concyclic