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:

- 1 radian = 1 degree . \(\frac{\pi}{180^{\circ}}\)
- 1 degree = 1 radian . \(\frac{180^{\circ}}{\pi}\)

Two arcs are congruent if they have the same measure and are arcs of the same circle or of congruent circles.

**Congruent arcs**have congruent chords and congruent central angles in the same circle or in congruent circles.*Theorem:*In the same circle or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.- \(\overset{ \huge\frown}{AB} \cong \overset{ \huge\frown}{CD}\) if and only if \(\overline{AB} \cong \overline{CD}\)

- \(\overset{ \huge\frown}{AB}\) and \(\overset{ \huge\frown}{CD}\) are not congruent because the arcs are not on congruent circles.

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.

- AE is the perpendicular bisector to BC, so AE is the diameter of the circle.

*Theorem: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.*

- AE is the diameter, so \(\overline{BD} \perp \overline{DC}\) and \(\overset{ \huge\frown}{BE} \cong \overset{ \huge\frown}{EC}\).

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.

- \(m \angle BCE = \frac{1}{2}\) of measure of arc intercepted by the chord = \(\frac{1}{2}\) the minor arc
- \(m \angle BCD = \frac{1}{2}\) of measure of arc intercepted by the chord = \(\frac{1}{2}\) the major 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.

- In the circle to the right:\(m\angle DPB = \frac{1}{2}(m\overset{ \huge\frown}{DB} + m\overset{ \huge\frown}{CE})\).
- Since ∠DPB and ∠CBE are vertical angles, m ∠DPB = m ∠CBE

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

- Example: For the diagram on the very left, \(m \angle EDF = \frac{m\overset{ \huge\frown}{EF} - m\overset{ \huge\frown}{GH}}{2}\)