Definition A cyclic quadrilateral is a quadrilateral in a circle where all four vertices of quadrilateral lie on the circumference of the circle.

• The sum of opposite interior angles in a cyclic quadrilateral is $$180^\circ$$.

 Example The diagram below shows a cyclic quadrilateral $$PQRS$$ and a straight line $$RST$$. Calculate the value of $$\angle PSR$$. First, calculate the value of $$y$$. \begin{aligned} \angle PQR + \angle PSR &= 180{^\circ} \\\\ 4y+ 2y&= 180{^\circ} \\\\6y&=180{^\circ} \\\\y&=30{^\circ}. \\\\\end{aligned} Thus, \begin{aligned}\angle PSR&=30^\circ\times2 \\\\&=60^\circ. \end{aligned}

• The sum of opposite interior angles in a cyclic quadrilateral is $$180^\circ$$.
 Example The diagram below shows a cyclic quadrilateral $$PQRS$$ and a straight line $$RST$$. Calculate the value of $$\angle PSR$$. First, calculate the value of $$y$$. \begin{aligned} \angle PQR + \angle PSR &= 180{^\circ} \\\\ 4y+ 2y&= 180{^\circ} \\\\6y&=180{^\circ} \\\\y&=30{^\circ}. \\\\\end{aligned} Thus, \begin{aligned}\angle PSR&=30^\circ\times2 \\\\&=60^\circ. \end{aligned}