Angles in a Circle
Keystone Geometry
Types of Angles
There are four different types of angles in any given circle. The
type of angle is determined by the location of the angles vertex.
1. In the Center of the Circle: Central Angle
2. On the Circle: Inscribed Angle
3. In the Circle: Interior Angle
4. Outside the Circle: Exterior Angle
* The measure of each angle is determined by the Intercepted Arc
Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if
each of the following conditions holds:
1. The endpoints of the arc lie on the angle.
2. All points of the arc, except the endpoints, are in the
interior of the angle.
3. Each side of the angle contains an endpoint of the arc.
Central Angle
Definition: An angle whose vertex lies on the center of
the circle.
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central
Angle
(of a circle)
* The measure of a central angle is equal to the measure of the
intercepted arc.
Measuring a Central Angle
The measure of a central angle is equal to
the measure of its intercepted arc.
123°
70°
28°
Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and
whose sides are chords of the circle (or one side tangent to
the circle).
Examples:
1
3
2
No!
Yes!
4
No!
Yes!
Measuring an Inscribed Angle
The measure of an inscribed angle is equal
to half the measure of its intercepted arc.
82°
Corollaries
If two inscribed angles intercept the same
arc, then the angles are congruent.
Corollary #2
An angle inscribed in a semicircle is a right angle.
Corollary #3
If a quadrilateral is inscribed in a circle, then
its opposite angles are supplementary.
88
82°
98°
92°
** Note: All of the
Inscribed Arcs
will add up to 360
Another Inscribed Angle
The measure of an angle formed by a chord
and a tangent is equal to half the measure of
the intercepted arc.
C
E
D
106°
B
A
Exterior Angles
An exterior angle is formed by two secants, two tangents,
or a secant and a tangent drawn from a point outside the
circle. The vertex lies outside of the circle.
Two secants
A secant and a tangent
Two tangents
Exterior Angle Theorem
The measure of the angle formed is equal to ½ the
difference of the intercepted arcs.
°
x -y
Ð1 =
2
°
°
x -y
Ð2 =
2
°
°
x -y
Ð3 =
2
°
Exterior Angle Theorem
In the given example: ÐACB is an exterior
Green° - Re d °
angle, therefore ÐACB =
.
2
265 ° - 95 °
ÐACB =
2
170°
ÐACB =
= 85°
2
Interior Angles
• An interior angle can be formed by two chords (or
two secants) that intersect inside of the circle.
• The measure of the angle formed is equal to ½ the
sum of the intercepted arcs.
BC + DE
mÐ1 =
2
Interior Angle Example
If BD = 40 ° and CE = 160 °
40 + 160 200
then mÐ1=
=
= 100 °
2
2
If BD = 28 ° and mÐ1 = 76 ° , find CE
28 + x
then 76 =
so 152 ° = 28 + x
2
so x=124 ° .
°