Consider a 60 degree angle inscribed in the center of a circle that corresponds to a certain arc. Now, consider another angle inside of the larger angle that cuts the larger angle in half. Turning these two angles and their corresponding arcs into triangles, we find that the second angle has half the arc length of the first triangle, for its arc is one of the sides of the bisected first arc. However, its angle is also half of that of the first arc. Thus, a smaller arc corresponds to a smaller angle, and a larger arc corresponds to a larger angle.