The GeoGebra app above illustrates the combined effect of rotating through the three angles in a triangle. Angle A rotates the bottom side of the triangle to its left side; angle B rotates the left side to the right side; and angle C rotates the right side to the bottom side. Note that the bottom side will then be facing in the opposite direction--the combined effect of the 180 degree rotation.
To make the argument stronger, consider the effect of the three composed rotations on the dashed line through the bottom side of the triangle. Angle A rotates it to the dashed line through the left side of the triangle. Now, the second rotation occurs about a different vertex, but if we ignore the translation and focus only on the orientation of the line, angle B rotates the line just as it would anywhere else in the plane; likewise with angle C. The composition of all three angles has the effect of flipping the line over (rotating it 180 degrees).
This argument fits with an expanded view of the Erlangen program, in which we consider geometric properties, such as angle rotation, left invariant under various transformations (such as translation). The article linked below elaborates on related arguments.