Question

Prove that if a pair of opposite angles of a quadrilateral are right, then a circle can be drawn through all four of its vertices.

Answer 1

Consider the figure below.

Here, ABCD is a quadrilateral in which ∠A = ∠C = 90°.

We know that angle in a semi circle is a right angle.

Suppose a circle is drawn such that it passes through vertex A.

As ∠BAD = 90°, BD acts like the diametre of this circle.

Similarly, suppose a circle is drawn such that it passes through vertex C.

As ∠BCD = 90°, BD acts like the diametre of this circle.

Thus, BD is the diametre of the circle passing through vertices A and C.

As the end points of the diametre lie on the circle, all the four vertices of quadrilateral ABCD lie on the circle.

Hence, if a pair of opposite angles of a quadrilateral are right then a circle can be drawn through all four of its vertices.