Final answer:
When an angle's vertex is inside but not at the center of a circle, the measure of the angle is the average of the intercepted arcs. This corresponds with the concept of rotation angle in relation to a circle's geometry.
Step-by-step explanation:
When an angle has a vertex inside the circle but NOT in the center nor inscribed, then you take the average of the intercepted arcs. This concept is related to circle geometry where angles formed by lines intersecting a circle can determine arc lengths. In this particular situation, the angle is known as an angle formed by a secant and a tangent or by two secants intersecting inside the circle, and its measure is equal to half the sum of the arc lengths it intercepts.
The rotation angle is intimately connected to the concept of arc length as it relates to the movement around a circular path. The rotation angle is the ratio of the arc length to the radius of curvature on a circular path. This relationship underpins many principles within circle geometry.