Math Terms: The Definition of an Angle

A Guide to Angles in the Study of Mathematics

Teacher helping boy draw angle on blackboard using protractor, rear view
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Angles are an integral facet in the study of mathematics, particularly geometry. Angles are formed by two rays (or lines) that begin at the same point or share the same endpoint. The point at which the two rays meet (intersect) is called the vertex. The angle measures the amount of turn between the two arms or sides of an angle and is usually measured in degrees or radians. 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" is derived from the Latin word "angulus," meaning "corner" and 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 measure exactly 90 degrees are called right angles. Angles that measure 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 measure greater than 90 degrees but less than 180 degrees are called obtuse angles. Angles that are larger than a straight angle but less than one 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 example, a typical rooftop is formed using an obtuse angle. The rays span out to accommodate the width of the house, with the apex located at the centerline of the house and the open end of the angle facing downward. The angle chosen must be sufficient to allow the water to flow off the roof easily but not so close to 180 degrees that the surface would be flat enough to allow water to pool.

If the roof were constructed at a 90-degree angle (again, with the apex at the centerline and the angle opening outward and facing down) the house would likely have a much narrower footprint. As the measurement of the angle decreases, so too does the space between the rays.

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 one another. 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 ">" and "<" shapes. Those would be angles "C" and "D." Both C and D would share the same degrees, as 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."