An angle has a magnitude and a direction.
Without the ability to freely specify direction, it is impossible to measure angles consistently.
Angle[] is pretend and I don't use it.
My system is compatible with all relevant theorems, from Euclid to Gauss.
It will take me some time to describe it and show that it is one and the same thing as the way we have already agreed to measure, and to flesh out the tools. To demonstrate a step, I must first solve all of the problems which I will use in the demonstration. Hence, a great deal of work near the end, and on largely irrelevant mechanics (pure logic) in order to reach back and start firmly at the beginning. This is a difficulty peculiar to working with computers.
Please let me know if there are particular tools which would be of use to you. I may already have one, and if not I am always open to challenges! Leave a post here:
http://www.geogebratube.org/material/show/id/91343