Notes
cyclic quadrilateral

Angles in a Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle.

An important result on cyclic quadrilaterals is that the interior angles at two opposite vertices add up to $180^\circ$. So in the above diagram we have:

One way to show this is to consider the following diagram, in which all the vertices of the quadrilateral have been joined to the centre of the circle.

All the new lines are radii of the circle and so are the same length, so all the triangles are isosceles. Using the fact that the sum of the interior angles of a quadrilateral is $360^\circ$ we have that:

and so

But the sum $a + b + c + d$ is also what is obtained by adding together the interior angles at two opposite vertices.