Arguing Angles

I'm working with a group of other Geometry teachers in my school trying to plan lessons together. Monday afternoon we got into a heated discussion. The topic? How to define the major arc of a circle.

We all agreed on the textbook definition of a minor arc: if the measure of the corresponding central angle is less than 180 degrees, then it is a minor arc. But the dispute was about the definition for a major arc.

My colleagues wanted to use parallel language, saying it is the arc for a central angle greater than 180 degrees. But our text has never discussed what it might mean to be greater than 180 degrees. In fact, were we to allow angle measures greater than 180 degrees, then we lose the definitions of "interior" and "exterior" of angles, and consequently a multitude other relationships become ambiguous or wrong.

I'd rather go with our textbook's definition of major arc, which uses the idea of "not interior to the central angle." This maintains the idea of angle being less than (or equal to) 180 degrees, and preserves definitions of  interior. But my colleagues say this is too complicated, and our students can't handle it.

My colleagues argued that in Trigonometry we regularly use angles much greater than 180 degrees and much less than 0. But I pointed out that in Trigonometry, in order to go beyond 90 degrees, we must view angles as rotations, while in Geometry they are static. This didn't persuade them.

We regularly have discussions about rigor, and holding our students to high expectations. How can we on one hand water down content and on the other hand say we are rigorously holding high expectations?