The angle between two arcs
Suppose that we want to define the signed angle between arcs PA and PB of curves meeting at a point P. Provided the curves have tangents at P, the sensible way to define the angle
Take a point Q on PA. As Q approaches P along the arc PA, R will approach P along arc PB.
In the limit, PQ approaches the tangent ray to arc PA at P,
In the picture, the curves and corresponding tangent rays are red and green.
To see this in the CabriJava box, drag the point Q towards P 
The Invariance Theorem
Let C be a circle with centre O.
Suppose that two arcs meet in a point P, We consider inversion with respect to C. The inverses of the arcs will be arcs meeting at P' (the inverse of P).
It is a remarkable fact that the angle between the inverse arcs The Invariance Theorem Inversion in a circle preserves the size of any angle, but reverses the sense. The CabriJava box on the right illustrates the Theorem, and suggests the proof!
If you drag Q towards P (it stays on the upper green arc),
The inverses are respectively Q' and R',
Of course, the rays PQ, PR, P'Q' and P'R' tend towards


Proof of the Invariance Theorem
Main inversive page 