An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. The other end points than the vertex, A and C define the intercepted arc ⌢AC of the circle. The measure of ⌢AC is the measure of its central angle.

An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle.

In a circle or congruent circles, congruent arcs are arcs with equal measures. 23. Angle and Arc Relationships in the Circle. To further clarify the definition of congruent arcs, consider the concentric circles (having the same center) in. Figure 6.10.

Specifically, given a chain of six circles all tangent to a seventh circle and each tangent to its two neighbors, the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same point.

An angle of a circle is an angle that is formed between the radii, chords, or tangents of a circle. We saw different types of angles in the “Angles” section, but in the case of a circle, there, basically, are four types of angles. These are central, inscribed, interior, and exterior angles.

Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc. The following two theorems directly follow from Theorem 70. Theorem 71: If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure.

The measure of an arc refers to the arc length divided by the radius of the circle. The arc measure equals the corresponding central angle measure, in radians. That’s why radians are natural: a central angle of one radian will span an arc exactly one radius long.

If an angle is a central angle, then its measure is equal to the measure of the intercepted arc.