# Angles in a Cyclic Quadrilateral A **cyclic quadrilateral** is a [[quadrilateral]] whose [[vertices]] lie on a circle. [[!include cyclic quadrilateral SVG]] 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: $$ a + c = 180^\circ, \qquad b + d = 180^\circ $$ 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. [[!include cyclic quadrilateral proof SVG]] 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 polygon|sum of the interior angles of a quadrilateral]] is $360^\circ$ we have that: $$ 2a + 2b + 2c + 2d = 360^\circ $$ and so $$ a + b + c + d = 180^\circ. $$ But the sum $a + b + c + d$ is also what is obtained by adding together the interior angles at two opposite vertices. [[!redirects angles in a cyclic quadrilateral]] [[!redirects opposite angles in a cyclic quadrilateral]] [[!redirects cyclic quadrilaterals]]