Inscribed Angles

 An inscribed angle is formed when two secant lines of a circle intersect on the circle. An inscribed angle is said to intersect an arc on the circle. The arc is the portion of the circle that is in the interior of the angle. The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle.

The inscribed angle theorem states that an angle ? inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. Therefore, the angle does not change as its apex is moved to different positions on the circle. The inscribed angle is only defined for points on the major arc. In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle.

Typically, it is easiest to think of an inscribed angle as being defined by two chords of the circle sharing an endpoint. An inscribed angle is said to intersect an arc on the circle. The arc is the portion of the circle that is in the interior of the angle. The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle.

This single property has a number of consequences within the circle. For example, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal. It also allows one to prove that the opposite angles of a cyclic quadrilateral are supplementary.

Whereas central angles are formed by radii, inscribed angles are formed by chords. The measure of an inscribed angle is half the measure of its intercepted arc. Inscribed angles subtended by the same arc of a circle are equal. Inscribed angles subtended by the same arc of a circle are equal. An angle inscribed in a circle has half the angle measure of the corresponding central angle