Many angles can be formed by parts of circles, and we can find their measures based on intercepted arcs.
Arc: A section of a circle.
Congruent Arcs: Arcs are congruent if their central angles are congruent.
Radians: A way of expressing angle measure based on arc length.
Inscribed Angle: An angle where the vertex is on the circle’s circumference and the sides contain chords.
An arc can be measured in degrees or in radians.
Radians are another way to express angle measure and are related to degrees. The radian is the ratio of the arc length and the radius. To convert between radians and degrees:
Two arcs are congruent if they have the same measure and are arcs of the same circle or of congruent circles.
If two arcs have the same central angle measure but are not on congruent circles, then the arcs are not congruent
The point E and any line, segment, or ray that contains E bisects \(\overset{ \huge\frown}{BEC}\) and causes \(\overset{ \huge\frown}{BE} \cong \overset{ \huge\frown}{CE}\)
Theorem: The perpendicular bisector of a chord is also a diameter.
Theorem: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.
An inscribed angle is formed by two chords and is one example of an angle on a circle.
The angle formed by a chord and a tangent of a circle is another type of angle on a circle.
Theorem: The measure of an angle formed by a chord and a tangent that intersect on the circle is half the measure of the intercepted arc.
Any angle with its vertex on a circle will be half the measure of the intercepted arc!
An angle inside a circle is formed by two intersecting chords. The vertex is not at the center, but it is still inside the circle
Theorem: The measure of the angle formed by two chords that intersect inside a circle is the average of the measure of the intercepted arcs.
An angle outside a circle is formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.
Theorem: The measure of an angle outside the circle and is equal to half the difference of the measures of the intercepted arcs