Opposite Angles in a Cyclic Quadrilateral Add Up to π Radians

Consider a cyclic quadrilateral: all four of its vertices lie on a circle. Join the centre of the circle to each vertex. This creates four isosceles triangles, each made from two radii and one side of the quadrilateral.

Label the angles of the quadrilateral at the circumference by w, x, y and z. In each isosceles triangle, the angles at the base are equal, so the angle at the centre is:

π − 2w, π − 2x, π − 2y, π − 2z.

These four central angles meet at a point, so together they make one full turn:

(π − 2w) + (π − 2x) + (π − 2y) + (π − 2z) = 2π.

Rearranging gives:

−2w − 2x − 2y − 2z + 4π = 2π

so

2w + 2x + 2y + 2z = 2π

and therefore

w + x + y + z = π.

From this, the opposite-angle relations in the quadrilateral follow directly:

w + z = π − (x + y),
z + y = π − (w + x).

So in any cyclic quadrilateral, each pair of opposite angles adds up to π radians.