What does the following notation mean in geometry angles? What does the following notation mean in geometry angles?

 A: The only time I have seen this notation it meant arc, namely $\smile\!\!AB$ would denote arc $AB$. In a similar context I have usually seen $$\stackrel{\Huge\frown}{ABC},$$
using three points to avoid ambiguity (that could also be solved by using directed arcs, so that $\stackrel{\LARGE\frown}{AB}$ and $\stackrel{\LARGE\frown}{BA}$ are different). Of course, this notation has the drawback when handwriting, as $\overline{ABC}$ could look very similar. Also, with typed text $\smile\!\!AB$ fits nicely into a line.
In the context of angles and circles, it probably means the length of the arc expressed in terms of the circle radius, which happens to be the measure of angle $\angle AOB$ in radians where $O$ is the center of the circle (as suggested by @lulu). It is no surprise, because radians, the (non-) unit of angle measurement, is expressed in terms of arc length. This guess is compatible with the theorem that passage is describing:

where the red angle is equal to the sum of light blue and light green angles, which are equal to half of the dark blue and dark green angles.
I hope this helps $\ddot\smile$
