There are various **types of angles**taking into consideration several criteria to differentiate them, for example they can be distinguished through their measurement, by the position they occupy and also according to the sum with other angles.

Usually an angle is defined as the opening included between two rays with a common origin, called the *vertex* of the angle. The *amplitude* of the aperture is the measure of the angle, which is often given in degrees or radians.

A degree is equal to one of the 360 parts into which a circumference can be divided. If the circumference is divided into two equal parts, each one is equivalent to 180 degrees or 180º, if instead it is divided into four equal parts, each one will be 90º and so on. This system is called *sexagesimal*.

Radians is another widely used measure, which consists of taking a circumference and measuring the angle between two of its radii, whose length is “r” and with the vertex in the center of the circle. In this way, the arc «s» between said radii is also asserted «r» and the subtended angle is then 1 radian or 1 rad and is equivalent to 57.3º degrees.

The instrument for measuring angles is the protractor. To take a measurement, the center of the protractor is made to coincide with the vertex of the angle and one of its sides with the 0º line of the protractor. The other side coincides with the measure of the angle, which is read on the scale.

**Types of angles according to their measure**

One of the most frequent ways of referring to angles is to name them according to their measure, although in some cases an angle can belong to more than one of the categories described below.

**null angle**

The one whose measure is 0º or 0 rad, that is, the two rays have a zero opening.

**Acute angle**

The measure of an acute angle is between 0 and 90º or between 0 and π/2 radians. For example, angles of 30º, 45º and 60º, which are part of the notable angles, are all acute angles.

**Right angle**

It is the one that measures exactly 90º (π/2 radians), this means that the half lines that define it are perpendicular to each other. The internal angles of a square or a rectangle are right angles, and the one formed between the legs of a right triangle is also a right angle.

**Obtuse angle**

It is an angle greater than 90º or π/2 radians.

**flat angle**

It measures exactly 180º, equivalent to π radians. When a vector magnitude is opposite to another, they form an angle of 180º with each other, for example the speed of a mobile moving in a straight line and the deceleration it experiences when it is stopping.

**convex angle**

Whenever an angle measures less than 180º it is a convex angle. An acute angle can be convex, as can one of 90º and those obtuse angles whose measure is between 90º and 180º. More examples of convex angles are:

**concave angle**

It is the one that measures more than 180º, such as 225º or 270º, the latter equivalent to three quarters of the circumference.

**Full angle or perigonal**

Its measure is 360º or 2π radians. It means that the two half lines that make it up coincide again, but unlike the zero angle, in this case a complete turn has been made.

**Types of angles according to the position of their sides**

In many figures and geometric structures there is more than one angle and that is why it is convenient to have a criterion to compare the sides of one with respect to another. In this way you have:

**consecutive angles**

Consecutive angles are next to each other, so they have a common side and vertex.

**adjacent angles**

Adjacent angles have a side and a vertex in common, that is, they lie next to each other. But unlike consecutive angles, in adjacent angles the remaining sides are opposite rays, therefore the two angles add up to 180º.

**Opposite angles by the vertex**

The angles opposite by the vertex have the vertex in common, and their sides extend opposite each other, from one of the angles to the other. In this way, the angles opposite by the vertex have the same measure.

The following figure shows 4 angles, denoted with Greek letters. The blue angles are α and β, and as can be seen, they are acute angles and opposite by the vertex. On the other hand, the angles γ and δ are obtuse angles and they are also opposite by the vertex.

**Types of angles according to the sum of their measures**

Some calculations, especially in trigonometry, are greatly simplified by observing if the sum of the measures of two angles is that of one of the notable angles, such as the right angle (90º) or the straight angle (180º). Accordingly, there are:

**complementary angles**

Those angles whose sum is equal to 90º are complementary. For example, the internal acute angles of a right triangle are complementary, since the sum of its three internal angles is equal to 180º.

Since one of the internal angles of the right triangle already measures 90º, the sum of the other two is also equal to 90º.

**supplementary angles**

They are those angles whose sum is equal to 180º. For example, the angles α and β shown in the upper figure.

Examples of notable angles that are also supplementary are:

120º and 60º 135º and 45º

**References**

Alexander, D. 2013. Geometry. 5th. Edition. Cengage Learning. Baldor. 1983. Plane and Space Geometry and Trigonometry. Cultural Homeland Group. EA 2003. Elements of geometry: with exercises and compass geometry. University of Medellin. Geometry 1st ESO. Angles in the circumference. Retrieved from: edu.xunta.es. Rich, B. Geometry. 1991. Schaum series. 2nd. Edition. McGraw Hill.