Angles in a Circle

Central Angles

Definition: Central Angle

Let $k(O;r)$ be a circle.

A central angle is an angle whose vertex is $O$ and whose rays are radii of $k$.

Arcs

Definition: Arc

An arc of a circle is a part of the circle bound between a central angle.

It is obvious that each central angle corresponds to a single arc and that each arc corresponds to a single central angle. Hence, we often use the terms “arc” and “central angle” interchangeably.

Inscribed Angles

Definition: Inscribed Angle

Let $k(O;r)$ be a circle.

An inscribed angle is an angle whose vertex lies on $k$ and whose rays intersect $k$ at differents point from the vertex.

Definition: Corresponding Arc

The corresponding arc of $\angle ABC$ is thearc$\stackrel{⌢}{\mathrm{AC}}$ which does not pass through $B$.

Definition: Corresponding Chord

The corresponding chord of $\angle ABC$ is the chord which corresponds to $\angle ABC$‘s corresponding arc.

Definition: Corresponding Central Angle

The corresponding central angle of $\angle ABC$ is the central angle $\angle AOC$ whose correspondingarcis the same as $\angle ABC$‘s corresponding arc.

Properties

Theorem: Measure of Inscribed Angles

Let $k(O;r)$ be a circle.

The measure of an inscribed angle is always half the measure of its corresponding central angle.

$$\angle ABC=\frac{1}{2}\angle AOC$$

PROOF

TODO

TIP

This means that inscribed angles which have the same correspondingarc/ chord / central angle are equal.

Theorem: Right Inscribed Angles

An inscribed angle has a measure of $90^{°}$ ($\frac{\pi }{2}\mathrm{rad}$) if and only if its corresponding chord is a diameter.

PROOF

TODO

Tangent Chord Angles

Definition: Tangent Chord Angle

Let $k(O;r)$ be a circle.

A tangent chord angle or peripheral angle is an angle whose vertex lies on $k$, whose one ray is a chord and whose other ray lies on a tangent line to $k$.

Definition: Corresponding Arc

The corresponding arc of a tangent chord angle is thearcwhich lies in its interior.

Definition: Corresponding Central Angle

The corresponding central angle of a tangent chord angle is the central angle whose corresponding arc is the same as the corresponding arc of the tangent chord angle.

Properties

Theorem: Measure of Tangent Chord Angles

The measure of a tangent chord angle is always half of its corresponding central angle.

$$\phi =\frac{1}{2}\angle AOB$$

PROOF

TODO

TIP

This means that all tangent chord angles with the same corresponding central angle are equal.

Interior Angles

Definition: Interior Angle

Let $k(O;r)$ be a circle.

An interior angle is an angle whose vertex lies inside of $k$ and whose rays intersect $k$.

Definition: Corresponding Arcs

The corresponding arcs of an interior angle are the arcs enclosed between its rays and the extensions of those rays.

Definition: Corresponding Central Angles

The corresponding central angles of an interior angle are the central angles with the same corresponding arcs as the interior angle.

$$\angle AOC,\angle A^{\prime }O{C}^{\prime }$$

Properties

Theorem: Measure of Interior Angles

The measure of an interior angle is always half of the sum of the measures of its corresponding central angles.

$$\angle ABC=\frac{1}{2}(\angle AOC+\angle A^{\prime }O{C}^{\prime })$$

PROOF

TODO

Exterior Angles

Definition: Exterior Angle

Let $k(O;r)$ be a circle.

An exterior angle is an angle whose vertex lies outside of $k$ but whose rays intersect $k$.

Definition: Corresponding Arcs

The corresponding arcs of an exterior angle are the arcs between the chords cut off by the angle’s rays.

Definition: Corresponding Central Angles

The corresponding central angles of an exterior angle are the corresponding central angles of its corresponding arcs.

Properties

Theorem: Measure of Exterior Angles

The measure of an exterior angle is always half of the difference between the measures of its corresponding central angles.

$$\angle ABC=\frac{1}{2}(\angle AOC-\angle A^{\prime }O{C}^{\prime })$$

PROOF

TODO