# Angles: type of angles, measure and operations

We say that an angle is the aperture that exists between two straight lines (or line segments) that intersect at a point called the vertex of the angle.

In this figure we can observe the aperture created by both lines (symbolized by the dotted curve) and this represents the angle that these lines form.

## Types of angles

We will observe that there are different types of angles. We define them next:

• Right angle: the angle formed by two perpendicular lines.

• Acute angle: an angle smaller than a right angle.

• Straight (or Flat) angle: it is the angle formed by two lines that form a single straight line.

• Obtuse angle: it is an angle smaller than an straight angle but larger than a right angle.

• Full angle: the angle formed by two superimposed lines.

• Reflex angle: an angle larger than an obtuse angle but smaller than a full angle.

## Measurement of angles

We measure angles with degrees and this is symbolized by the sign $^\circ$ (for example: we express $93$ degrees as $93^\circ$).

To establish this measurement we divide a full angle in $360$ degrees, and from this definition we can know what one degree measures.

To understand it better let's remember that a full angle is the angle formed by two lines that are superimposed:

A full angle is an angle of $360$ degrees

Once this measure is established, we can see:

• A right angle measures $90^\circ$.
• An acute angle measures between $0^\circ$ and $90^\circ$.
• A straight angle measures $180^\circ$.
• An obtuse angle measures between $90^\circ$ and $180^\circ$.
• A full angle measures $360^\circ$.
• A reflex angle measures between $180^\circ$ and $360^\circ$.

and also we can see:

• Two right angles form a straight one ($90^\circ+90^\circ = 180^\circ$).
• Two straight angles form a full one ($180^\circ+180^\circ = 360^\circ$).
• Four right angles form a full one ($90^\circ+90^\circ+90^\circ+90^\circ = 360^\circ$).

## Sum of angles

As we can see, we can add angles up, but what happens if the sum is greater than an angle of $360$ degrees?

Well, we have defined the angles from the angle of $0^\circ$ up to $360^\circ$ and we may notice that the relative position of two straight lines in positions of $0^\circ$ and of $360^\circ$ are equivalent:

This means that if by adding two angles together we get a total which is greater than $360^\circ$, we can look for an angle between $0^\circ$ and $360^\circ$ equivalent to the sum of these two angles.

For example,

If we add an angle of $90^\circ$ and one of $360^\circ$, we obtain one of $450^\circ$, which is equivalent to one of $90^\circ$:

plus =

Methodically, if we add angles that come to a total greater than $360^\circ$, to obtain the equivalent angle placed between $0^\circ$ and $360^\circ$ we have to successively subtract $360^\circ$ until finding an angle with a total of a maximum of $360^\circ$.

Let's add the angles $90^\circ, 180^\circ, 66^\circ, 25^\circ, 300^\circ, 21^\circ$ and $80^\circ$:

$$90^\circ+180^\circ+66^\circ+25^\circ+300^\circ+21^\circ+80^\circ = 762^\circ$$

and now let's subtract $360^\circ$ successively until we find an angle not bigger than $360^\circ$:

$$762^\circ - 360^\circ = 402^\circ$$ $$402^\circ - 360^\circ = 42^\circ$$

Consequently, the sum of all the previous angles turns out to be an angle of $42$ degrees.

## Subtraction of angles

In the same way as we have defined the sum of angles, let's define the angle subtraction.

For example,

A straight angle minus a right angle turns out to be a right angle:

minus =

Let's see what happens if, substracting several angles, we obtain a negative value.

As with the sum, the value of a negative angle is equivalent to the value of an angle between $0^\circ$ and $360^\circ$ and, in order to find this value, all we have to do is adding $360^\circ$ successively until we get a value within the desired range (between $0^\circ$ and $360^\circ$)

Let's do the subtraction of the angles $0^\circ, 25^\circ, 36^\circ, 152^\circ, 180^\circ, 36^\circ$ and $90^\circ$:

$$0^\circ-25^\circ-36-152^\circ-180^\circ-36^\circ-90^\circ =-519^\circ$$

and successively, we will be adding up $360^\circ$ until we reach a value between $0^\circ$ and $360^\circ$:

$$-519^\circ + 360^\circ =-159^\circ$$

$$-159^\circ + 360^\circ = 201^\circ$$

Consequently, the subtraction of all the previous angles turns out to be an angle of $201$ degrees.

## Angle bisector

We will say that the angle bisector of an angle formed by two lines is the angle formed by a third line that divides the original angle in two identical angles:

In this drawing we can see that the red line divides the angle formed by the other two lines in two halves.

To calculate the angle formed by the bisector, we only have to divide the value of the initial angle by two.

Given an angle of $42^\circ$, find the bisector angle.

We divide $42$ by $2$ and find that:$$\dfrac{42^\circ}{2}=21^\circ$$

Consequently, the bisector line has an angle of $21$ degrees.