The Definition of an Angle

Types of Angles in Math Terms

Teacher helping boy draw angle on blackboard using protractor, rear view
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In mathematics, particularly geometry, angles are formed by two rays (or lines) that begin at the same point or share the same endpoint. The angle measures the amount of turn between the two arms or sides of an angle and is usually measured in degrees or radians. Where the two rays intersect or meet is called the vertex. 

An angle is defined by its measure (for example, degrees) and is not dependent upon the lengths of the sides of the angle.

History of the Word

The word "angle" comes from the Latin word angulus, meaning "corner." It is related to the Greek word ankylοs meaning "crooked, curved," and the English word "ankle." Both Greek and English words come from the Proto-Indo-European root word "ank-" meaning "to bend" or "bow." 

Types of Angles

Angles that are exactly 90 degrees are called right angles. Angles less than 90 degrees are called acute angles. An angle that is exactly 180 degrees is called a straight angle (this appears as a straight line). Angles that are greater than 90 degrees and less than 180 degrees are called obtuse angles. Angles that are larger than a straight angle but less than 1 turn (between 180 degrees and 360 degrees) are called reflex angles. An angle that is 360 degrees, or equal to one full turn, is called a full angle or complete angle.

For an example of an obtuse angle, the angle of a typical house rooftop is often formed at an obtuse angle. An obtuse angle is greater than 90 degrees since water would pool on the roof (if it was 90 degrees) or if the roof did not have a downward angle for water to flow. 

Naming an Angle

Angles are usually named using alphabet letters to identify the different parts of the angle: the vertex and each of the rays. For example, angle BAC, identifies an angle with "A" as the vertex. It is enclosed by the rays, "B" and "C." Sometimes, to simplify the naming of the angle, it is simply called "angle A."

Vertical and Adjacent Angles

When two straight lines intersect at a point, four angles are formed, for example, "A," "B," "C," and "D" angles.

A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles. The opposite angles are mirror images of each other. The degree of angles will be the same. Those pairs are named first.  Since those angles have the same measure of degrees, those angles are considered equal or congruent. 

For example, pretend that the letter "X" is an example of those four angles. The top part of the "X" forms a "v" shape, that would be named "angle A." The degree of that angle is exactly the same as the bottom part of the X, which forms a "^" shape, and that would be called "angle B." Likewise, the two sides of the "X" form a ">" and an "<" shape. Those would be angles "C" and "D." Both C and D would share the same degrees, they are opposite angles and are congruent.

In this same example, "angle A" and "angle C" and are adjacent to each other, they share an arm or side. Also, in this example, the angles are supplementary, which mean that each of the two angles combined equals 180 degrees (one of those straight lines that intersected to form the four angles). The same can be said of "angle A" and "angle D."