Geometry

Arcs and Angles

Big Picture

Many angles can be formed by parts of circles, and we can find their measures based on intercepted arcs.

Key Terms

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.

Measure of an Arc

An arc can be measured in degrees or in radians.

Degrees and 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 . \(\frac{180^{\circ}}{\pi}\)

Congruent Arcs

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 \(\overline{AB} \cong \overline{CD}\)
Congruent Arcs

If two arcs have the same central angle measure but are not on congruent circles, then the arcs are not congruent

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Geometry

Arcs and Angles Cont.

Bisecting Arcs

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 \(\overset{ \huge\frown}{BE} \cong \overset{ \huge\frown}{EC}\).
Bisecting Arcs

Angles in a Circle

Angles on a Circle

An inscribed angle is formed by two chords and is one example of an angle on a circle.

inscribed angle

The angle formed by a chord and a tangent of a circle is another type of angle on a circle.

Angles on a Circle

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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!

Angles Inside a Circle

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
Angles Inside a Circle

Angles Outside a Circle

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.

Angles Outside a 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, \(m \angle EDF = \frac{m\overset{ \huge\frown}{EF} - m\overset{ \huge\frown}{GH}}{2}\)