# Angle

Pronunciation: /ˈæŋ.gəl/ Explain

An angle is the rotation between two intersecting lines, ray or line segments. Figure 1: Various angles

The vertex of an angle is the point of intersection of the lines. 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.

Two angles are equiangular if the measures of the angles are the same.

### Article Index

 Vertex of an Angle legs of an Angle Equiangular Measure of an Angle Degree Radian Gradian Classes of Angles Acute Angle Right Angle Obtuse Angle Straight Angle Reflex Angle Full Angle Inscribed Angle Central Angle Angle of Rotation Complementary Angle Supplementary Angle Copy an Angle Angle Bisector Bisect an Angle Angle Addition Postulate

### 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. 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.

#### 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.

1. The degree measure of an angle that is 1/3 of a circle is: Click for answer.360 / 3 = 120
2. The degree measure of an angle that is 1/17 of a circle is: Click for answer.360 / 17 ≈ 21.17

 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.

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: eiθ = 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.

1. The radian measure of an angle that is 1/3 of a circle is: Click for answer.2·π / 3 ≈ 2.09
2. The radian measure of an angle that is 1/17 of a circle is: Click for answer.2·π / 17 ≈ 0.37

### 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. The class an angle belongs to is determined by its measure. Table 1 shows the classes of angles and their measures. Manipulative 3 also shows the classes of angles and their measures.

 Ex. Angle Measure Class Degrees Radians 0 < θ < 90 0 < θ < π/2 Acute angle θ = 90 θ = π/2 Right angle 90 < θ < 180 π/2 < θ < π Obtuse angle θ = 180 θ = π Straight angle 180 < θ < 360 π < θ < 2·π Reflex angle θ = 360 θ = 2·π Full angle

### 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

1. All inscribed angles are greater than what measure?
2. All inscribed angles are smaller than what measure?
3. How does the measure of the inscribed angle change when only the vertex is moved without moving across one of the endpoints?
4. How does the measure of the inscribed angle change when the vertex is moved across one of the endpoints?
5. If an angle is inscribed in the diameter of a circle, what is the measure of the inscribed 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.

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 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.

### 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 degrees. Manipulative 8 - Complementary Angles Created with GeoGebra.
Two angles are complementary if they produce a right angle when combined. 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. 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.

 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 non-collinear points A, B, C and a point D in the interior of ∠BAC, m∠BAD + m∠DAC = m∠BAC.

### Table of Figures

1. McAdams, David E.. All Math Words Dictionary, angle. 2nd Classroom edition 20150108-4799968. pg 15. Life is a Story Problem LLC. January 8, 2015. Buy the book

### Educator Resources

• Angles and Direction Experiment . NASA LaRC Office of Education. 2/18/2010. http://www.archive.org/details/NasaConnect-Wygtya-AnglesAndDirectionExperiment/index.htm.
• Angle Activity. NASA LaRC Office of Education. 2/18/2010. http://www.archive.org/details/NasaConnect-Eom-AngleActivity/index.htm.
• Geometry of Exploration - Eyes Over Mars. NASA LaRC Office of Education. 2/18/2010. http://www.archive.org/details/NasaConnect-GeometryOfExploration-EyesOverMars/index.htm.

McAdams, David E. Angle. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/a/angle.html.

### Revision History

4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/28/2018: Corrected spelling. (McAdams, David E.)