Central and Inscribed Angles

A central angle has its vertex at the center of the circle, and its measure equals the measure of the arc it cuts off. An inscribed angle has its vertex somewhere on the circle's rim, with both rays reaching out to chords. Slide that rim vertex anywhere else along the circle, as long as it stays on the major arc outside the two endpoints, and the angle doesn't change. That's because every inscribed angle on the same arc is tied to the same central angle sitting over that arc, and the relationship between them is fixed: the inscribed angle is always half the central angle.

The reason for the half comes from the triangles you can draw by connecting the center to the inscribed angle's vertex and to the arc's endpoints. Those triangles are isosceles, since two sides are radii, and working through their base angles shows the central angle splits into pieces that are each twice the matching piece of the inscribed angle. The result holds for any inscribed angle on that arc, which is why the vertex can slide freely without changing the angle.

This rule only works when the vertex sits on the circle itself. Move the vertex inside the circle, off-center, and the angle formed by two chords instead equals half the sum of the two arcs it and its vertical angle cut off, not half of one single arc. A useful extension: when the arc under an inscribed angle is a semicircle, the central angle is 180 degrees, so the inscribed angle is always 90 degrees, a fact that connects directly to right triangles inscribed in circles.