Angle
Pronunciation: /ˈæŋ gəl/ Explain
An angle is the
rotation
between two
intersecting
lines,
ray or
line segments^{[1]}.

Figure 1: Various angles 
The vertex of an angle is the point of
intersection
of the lines.^{[3 page 9]} In figure 1, the vertices are the points b, h, and i.
The legs, also called sides, of an angle are the lines, rays, or line segments that define the angle.^{[3 page 9]}
Two angles are equiangular if the measures of
the angles are the same.
Article Index
Measure of an Angle
The measure of an angle is made in terms of the measure of a full circle. The unit of
measure for an angle is degrees, radians or, in rare cases, gradians.
Degree
Click on the blue point and drag it to change the figure.
How many degrees in a full circle?
 Manipulative 1  Angle Measured in Degrees Created with GeoGebra. 

One degree is 1/360 of a circle.^{[3 page 10]}
Degree is the oldest unit of measure for
an angle. Degrees are denoted by a small circle (°) or the abbreviation
deg. By definition, a full circle is
360°. This means that an angle that is 1/4 of a circle is 360°/4, or 90°. See manipulative 1.
Click on the blue point in manipulative 1 and drag it to change the figure.
Understanding Check
Calculate the answer to the problem and write it down. Then click on the blue and yellow words to see the correct answer.
 The degree measure of an angle that is 1/3 of a circle is: Click for answer.360 / 3 = 120
 The degree measure of an angle that is 1/17 of a circle is: Click for answer.360 / 17 ≈ 21.17

Radian
Click on the blue point and drag it to change the figure.
How many radians is a full circle?
 Manipulative 2  Angle Measured in Radians Created with GeoGebra. 

When measuring angles, the radian is particularly useful.
A radian is defined as the angle made with an arc length of
1 on a
unit circle.
This means that the length of an arc of 1 radian is the same as the length of
the radius of the circle. See manipulative 2. There are
2π
radians in a full circle. An angle that is 1/5 of a circle is
2π/5 ≈ 1.26 radians. The abbreviation for radian is
rad. Click on the blue point in manipulative 1 and drag it
to change the figure.
One reason the measure of radians is so useful has to do with Euler's famous equation
that relates exponentiation with trigonometry using
complex numbers:
e^{iθ} = cos(θ) + i·sin(θ).
This equation only works if the angle θ
is measured in radians.
Understanding Check
Calculate the answer to the problem and write it down. Then click on the blue and yellow words to see the correct answer.
 The radian measure of an angle that is 1/3 of a circle is: Click for answer.2·π / 3 ≈ 2.09
 The radian measure of an angle that is 1/17 of a circle is: Click for answer.2·π / 17 ≈ 0.37

Gradian
A rarely used angle measure is gradians. A full
circle measures 400 gradians. The abbreviation for gradians is grad.
Classes of Angles
Click on the blue point and drag it to change the figure.
Can you find all 5 angle classes represented?
 Manipulative 3  Angle Classes Created with GeoGebra. 

For convenience in discussions of angles and trigonometry, angles are divided into classes.^{[3 page 10]}
The class an angle belongs to is determined by its measure. Table 1 shows the classes of
angles and their measures. Click on the blue point in manipulative 3 and drag it to change
the figure. Manipulative 3 also shows the classes of angles and their measures.

Inscribed Angle
Click on the blue points and drag them to change the figure.
What is the largest angle that can be inscribed in a circle?
 Manipulative 4  Angle Inscribed in a Circle Created with GeoGebra. 

An inscribed angle is an angle drawn inside a
circle.
Discovery
 All inscribed angles are greater than what measure?
Click for Answer
All inscribed angles are greater than zero. Move the
endpoints
of the legs of the angle close together, but away from the vertex. The closer the
endpoints are to each other, they closer the angle is to zero.
 All inscribed angles are smaller than what measure?
Click for Answer
All inscribed angles are smaller than a straight angle. Move
the endpoints of the legs of the angle close to each side of the vertex, but on
opposite sides of the vertex. The closer the endpoints are to the vertex,
they closer the angle is to a straight angle.
 How does the measure of the inscribed angle change when only the vertex is moved
without moving across one of the endpoints?
Click for Answer.
If the endpoints of the inscribed angle are left alone and the vertex
of the inscribed angle is moved, the measure of the angle does not change.
 How does the measure of the inscribed angle change when the vertex is moved
across one of the endpoints?
Click for Answer.
If the vertex is moved across one of the endpoints, the
previous angle and the new angle are complementary. The angles add up to a straight
angle.
 If an angle is inscribed in the diameter of a circle, what is the measure of the
inscribed angle?
Click for Answer.
The measure of an angle inscribed in the diameter of
a circle is a right angle.

Central Angle
Click on the blue points and drag them to change the figure.
What is the largest angle that can be centrally inscribed in a circle?
 Manipulative 5  Central Angle Created with GeoGebra. 

A central angle of a circle is an angle with the
vertex at the center of the circle and the other two points on the circumference
of the circle.^{[3 page 41]} Click on the blue points in manipulative 6 and drag them to change
the figure.
For an inscribed angle and a central angle with the same endpoints, the measure
of the inscribed angle is half the measure of the central angle. Click on the
blue points in manipulative 7 and drag them to change the figure.

Click on the blue points and drag them to change the figure.
What is the relationship between a central angle and an inscribed angle that share two endpoints?
 Manipulative 6  Central Angle and Inscribed Angle Created with GeoGebra. 

Angle of Rotation
Click on the blue dot and drag it to change the figure.
What angle of rotation places the object back in the same place?
 Manipulative 7  Angle of Rotation Created with GeoGebra. 

An Angle of Rotation
is the amount of
rotation
about a
center of rotation.
Click on the blue points in manipulative 8 and drag them to change the figure.

Complementary Angles
Click on the blue points and drag them to change the figure.
Note that the measure of angle α and the measure of angle &beta always add up to 90 degress.
 Manipulative 8  Complentary Angles Created with GeoGebra. 

Two angles are complementary if they produce a
right angle
when combined.^{[3, page 3]} This means that given angles α and
β, α and
β are complementary if
α + β = π/2.
Supplementary Angles
Two angles are supplementary if they produce a
straight line when combined.^{[3, page 3]} This means that given angles
α and β,
α and β
are supplementary if α + β = π.


Click on the blue points and drag them to change the figure.
Note that the measures of angle &alpha and &beta always add up to 180 degrees.
 Manipulative 9  Supplementary Angles Created with GeoGebra. 


Copying an Angle
An angle can be
copied using a
compass
and a straight edge.
Click on the blue points and drag them to change the figure. Click on the Step slider and drag it to see each step.
Review the steps for copying an angle, then try it on a piece of paper.

Manipulative 10  How to Copy an Angle Created with GeoGebra. 
Angle Bisector
Click on the blue points and drag them to change the figure.
What can you say about the angles on each side of the angle bisector?
 Manipulative 11  Angle Bisector Created with GeoGebra. 

An angle bisector is a line segment or ray that divides an angle into two congruent angles. For details on how to bisect an angle, see How to Bisect an Angle.

Angle Addition Postulate
Click on the blue points and drag them to change the figure.
What is the relationship between the three angles?

Manipulative 12  Angle Addition Postulate Created with GeoGebra. 
The angle addition postulate states that
adjacent
angles can be added together to form a larger angle. This is a postulate or
axiom,
meaning it is accepted as true without proof. The exact mathematical definition of the
Angle Addition Postulate is:
Given
noncollinear
points A, B, C and a point
D in the interior of ∠BAC, m∠BAD + m∠DAC = m∠BAC.
 Various angles
 90 degree angle
 1 radian
 Inscribed angles
 An angle inscribed in a circle
 An angle inscribed in the diameter of a circle
 Central angle of a circle.
 Relationship between central and inscribed angles
 Complementary Angles
 Supplementary Angles
 Copying an Angle
 Angle Bisector
 Bisecting an Angle
Educator Resources
Cite this article as:
McAdams, David E. Angle. 7/10/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/angle.html.
Image Credits
Revision History
6/14/2018: Removed broken links, changed Geogebra links to work with Geogebra 5, updated license, implemented new markup. (
McAdams, David E.)
12/26/2009: Added "References". (
McAdams, David E.)
10/28/2008: Changed manipulatives and some graphics to geogebra. (
McAdams, David E.)
9/19/2008: Changed heading 'Other Information' to 'More Information', dicitonary.com to more information (
McAdams, David E.)
7/7/2008: Corrected link errors. (
McAdams, David E.)
4/28/2008: Added keyword class to angle classifications. (
McAdams, David E.)
4/19/2008: Revised bisecting an angle table to reflect most common method. (
McAdams, David E.)
3/11/2008: Fixed various formatting and link errors. (
McAdams, David E.)
2/3/2008: Changed HTML entity angle to the word angle. (
McAdams, David E.)
8/17/2007: Initial version. (
McAdams, David E.)