What is the relationship between the angles in a cyclic quadrilateral?

In a cyclic quadrilateral, the opposite angles add up to 180 degrees.

A cyclic quadrilateral is a four-sided figure where all the vertices lie on the circumference of a circle. This special property leads to an interesting relationship between its angles. Specifically, if you take any pair of opposite angles in a cyclic quadrilateral and add them together, their sum will always be 180 degrees. This is known as the supplementary property of opposite angles in a cyclic quadrilateral.

To understand why this happens, let's delve a bit deeper. Imagine a cyclic quadrilateral ABCD inscribed in a circle. The opposite angles are ∠A and ∠C, and ∠B and ∠D. According to the inscribed angle theorem, an angle subtended by an arc at the circumference is half the angle subtended by the same arc at the centre. Therefore, the angles subtended by the arcs AB and CD at the centre of the circle add up to 360 degrees because they form a complete circle. Since each angle at the circumference is half of the central angle, the sum of the opposite angles at the circumference will be half of 360 degrees, which is 180 degrees.

This property is very useful in solving various geometric problems and proofs. For example, if you know three angles of a cyclic quadrilateral, you can easily find the fourth angle by subtracting the sum of the known opposite angle from 180 degrees. Understanding this relationship helps in analysing and solving problems involving cyclic quadrilaterals more efficiently.