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Imagine two right triangles with same angles and different lengths of their sides. Say you know lengths of sides for only one of those triangles. (call them a1, b1, c1). Can you infer something about sides of another triangle (a2, b2, c2)? Yes you can because they are similar and therefore a1/b1 = a2/b2. Now notice this depends only on one angle of the triangle (other one is 90 degrees, and you can infer the third). So you just defined one mapping point of some function f: angle --> number. Now you sit down and create the chart (i.e. function) for every possible angle. You just created one trigonometric function and now its possible to use knowledge about angles to find lengths of sides of triangles. I see no reason why you couldn't do this for every geometric figure. For example create mapping for all triangles (not just right ones). You would create a function like f: (angle1, angle2) --> numbers. And this seems like a very natural thing to discover, you just need the knowledge about similarity in plane geometry.
