Central Angles#

Definition: Central Angle

Let \(k(O; r)\) be a [[Circle#Circle|circle]].

A central angle is an [[Plane Angle|angle]] whose vertex is \(O\) and whose rays are [[Circle#Circle|radii]] of \(k\).

![[res/Central Angle.svg]]

Arcs#

Definition: Arc

An arc of a [[Circle]] is a part of the circle bound between a [[Angles#Central Angles|central angle]].

![[res/Arc.svg]]

It is obvious that each [[Angles#Central Angles|central angle]] corresponds to a single [[Angles#Arcs|arc]] and that each [[Angles#Arcs|arc]] corresponds to a single [[Angles#Central Angles|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#Circle|circle]].

An inscribed angle is an [[Plane Angle|angle]] whose vertex lies on \(k\) and whose rays intersect \(k\) at differents point from the vertex.

![[res/Inscribed Angle.svg]]

Definition: Corresponding Arc

The corresponding arc of \(\angle ABC\) is the[[Angles#Arcs|arc]]\(\overset{\frown}{AC}\) which does not pass through \(B\).

![[res/Corresponding Arc of Inscribed Angle.svg]]

Definition: Corresponding Chord

The corresponding chord of \(\angle ABC\) is the [[Chords|chord]] which corresponds to \(\angle ABC\)'s corresponding arc.

![[res/Corresponding Chord of Inscribed Angle.svg]]

Definition: Corresponding Central Angle

The corresponding central angle of \(\angle ABC\) is the [[Circle#Central Angles|central angle]] \(\angle AOC\) whose corresponding[[Angles#Arcs|arc]]is the same as \(\angle ABC\)'s corresponding arc.

![[res/Corresponding Central Angle of Inscribed Angle.svg]]

Properties#

Theorem: Measure of Inscribed Angles

Let \(k(O; r)\) be a [[Circle#Circle|circle]].

The measure of an [[Angles#Inscribed Angles|inscribed angle]] is always half the measure of its [[Angles#Inscribed Angles|corresponding central angle]].

\[ \angle ABC = \frac{1}{2}\angle AOC \]

![[res/Measure of Inscribed Angle.svg]]

Proof

TODO

Tip

This means that inscribed angles which have the same corresponding[[Angles#Arcs|arc]]/ chord / central angle are equal.

Theorem: Right Inscribed Angles

An [[Angles#Inscribed Angles|inscribed angle]] has a measure of \(90^\degree\) (\(\frac{\pi}{2} \mathrm{rad}\)) if and only if its [[Angles#Inscribed Angles|corresponding chord]] is a [[Circle|diameter]].

![[res/Right Inscribed Angle.svg]]

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 [[Plane Angle|angle]] whose vertex lies on \(k\), whose one ray is a [[Chords|chord]] and whose other ray lies on a [[Configurations of Lines and Circles|tangent line]] to \(k\).

![[res/Tangent Chord Angle.svg]]

Definition: Corresponding Arc

The corresponding arc of a [[Angles#Tangent Chord Angles|tangent chord angle]] is the[[Angles#Arcs|arc]]which lies in its interior.

![[res/Corresponding Arc of Tangent Chord Angle.svg]]

Definition: Corresponding Central Angle

The corresponding central angle of a [[Angles#Tangent Chord Angles|tangent chord angle]] is the [[Angles#Central Angles|central angle]] whose [[Angles#Central Angles|corresponding arc]] is the same as the [[Angles#Tangent Chord Angles|corresponding arc]] of the [[Angles#Tangent Chord Angles|tangent chord angle]].

![[res/Corresponding Central Angle of Tangent Chord Angle.svg]]

Properties#

Theorem: Measure of Tangent Chord Angles

The measure of a [[Angles#Tangent Chord Angles|tangent chord angle]] is always half of its [[Angles#Tangent Chord Angles|corresponding central angle]].

\[ \varphi = \frac{1}{2} \angle AOB \]

![[res/Measure of Tangent Chord Angle.svg]]

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#Circle|circle]].

An interior angle is an [[Plane Angle|angle]] whose vertex lies inside of \(k\) and whose rays intersect \(k\).

![[res/Interior Angle in a Circle.svg]]

Definition: Corresponding Arcs

The corresponding arcs of an [[Angles#Interior Angles|interior angle]] are the arcs enclosed between its rays and the extensions of those rays.

![[res/Corresponding Arcs of Interior Angle.svg]]

Definition: Corresponding Central Angles

The corresponding central angles of an [[Angles#Interior Angles|interior angle]] are the [[Angles#Central Angles|central angles]] with the same [[Angles#Central Angles|corresponding arcs]] as the interior angle.

\[ \angle AOC, \angle A'OC' \]

![[res/Corresponding Central Angles of Interior Angle.svg|Corresponding Central Angles of Interior Angle]]

Properties#

Theorem: Measure of Interior Angles

The measure of an [[Angles#Interior Angles|interior angle]] is always half of the sum of the measures of its [[Angles#Interior Angles|corresponding central angles]].

\[ \angle ABC = \frac{1}{2}(\angle AOC + \angle A'OC') \]

![[res/Corresponding Central Angles of Interior Angle.svg|Corresponding Central Angles of Interior Angle]]

Proof

TODO

Exterior Angles#

Definition: Exterior Angle

Let \(k(O; r)\) be a [[Circle#Circle|circle]].

An exterior angle is an [[Plane Angle|angle]] whose vertex lies outside of \(k\) but whose rays intersect \(k\).

![[res/Exterior Angle.svg]]

Definition: Corresponding Arcs

The corresponding arcs of an [[Angles#Exterior Angles|exterior angle]] are the arcs between the [[Chords]] cut off by the angle's rays.

![[res/Corresponding Arcs of Exterior Angle.svg]]

Definition: Corresponding Central Angles

The corresponding central angles of an [[Angles#Exterior Angles|exterior angle]] are the [[Angles#Central Angles|corresponding central angles]] of its [[Angles#Exterior Angles|corresponding arcs]].

![[res/Corresponding Central Angles of Exterior Angle.svg]]

Properties#

Theorem: Measure of Exterior Angles

The measure of an [[Angles#Exterior Angles|exterior angle]] is always half of the difference between the measures of its [[Angles#Exterior Angles|corresponding central angles]].

\[ \angle ABC = \frac{1}{2} (\angle AOC - \angle A'OC') \]

![[res/Corresponding Central Angles of Exterior Angle.svg]]

Proof

TODO