Cyclic quadrilaterals - Higher
A cyclic quadrilateral is a A quadrilateral is a shape with four straight sides and four angles. drawn inside a circle. Every vertex of the quadrilateral must touch the circumference of the circle.
The second shape is not a cyclic quadrilateral. One vertex does not touch the circumference.
The opposite angles in a cyclic quadrilateral add up to 180°.
\(a + c = 180^\circ\)
\(b + d = 180^\circ\)
Example
Calculate the angles \(a\) and \(b\).
The opposite angles in a cyclic quadrilateral add up to 180°.
\(b = 180^\circ - 140^\circ = 40^\circ\)
\(a = 180^\circ - 60^\circ = 120^\circ\)
Proof
Let angle CDE = \(x\) and angle EFC = \(y\).
The angle at the centre is twice the angle at the circumference.
Angle COE = \(2y\) and the reflex angle COE = \(2x\).
Angles around a point add up to 360°.
\(2x + 2y = 360^\circ\)
\(\frac{2x}{2} + \frac{2y}{2} = \frac{360^\circ}{2}\)
So \(x + y = 180^\circ\)