Sorry, I labelled the angles the same as your vertices.

This was not intentional.

Join all corners of the quadrilateral to the circle centre "o".

you now have 4 isosceles triangles with the angles at the circumference within each triangle equal.

Therefore 2A+2B+2C+2D =360 degrees.

A+B+C+D=180 degrees.

(A+B)+(C+D)=180 degrees.

(A+C)+(B+D)=180 degrees.