# Finding missing angles

## In a nutshell

Angles in a triangle add up to $180\degree$. Angles in a quadrilateral add up to $360\degree$. Using this information, you can calculate the sum of the internal angles in any polygon and find any missing angles.

## Angles in a triangle

All internal angles in a triangle add up to $180\degree$, regardless of the type of triangle.

A **right-angle **triangle has one angle which is exactly $90\degree$. This means the other two angles add up to $90\degree$.

In an **equilateral** triangle, all 3 angles are equal and are $60\degree$.

An **isosceles **triangle has two angles which are equal.

In a **scalene** triangle, all the angles are different.

Using this knowledge of angles in a triangle, you can work out any missing angle by subtracting the given angles from $180\degree$.

##### Example 1

*Find the missing angle*

*Add up the angles given:*

*$80+40=120$*

*Take away this number from $180$*:

$180-120=60$

*The missing angle is $\underline{60\degree}$.*

## Angles in a quadrilateral

A quadrilateral is a shape which has four sides, and can be regular or irregular. Angles in any quadrilateral add up to $360\degree$.

**Squares** and **rectangles** are made up of four right-angles. You will never have to work out missing angles in these two shapes, but they can be useful when trying to find missing angles in other polygons.

**Parallelograms **and** rhombuses **have two pairs of equal angles - the angles opposite each other are equal. In a rhombus a line can be drawn connecting one corner to the opposite corner and these two lines will intersect to create right-angles.

**Note:** Both shapes have two sets of parallel lines. It's important to remember this to apply the opposite angles rule.

**Trapeziums **have only one pair of parallel lines, where each line is a different length.

**Kites** have one pair of equal, opposite angles.

**Irregular quadrilaterals** are shapes which have 4 sides of different lengths.

##### Example 2

*If angle C is $140\degree$, what is angle B?*

*Angles given are two right angles and $140\degree$. Add these together;*

*$90+90+140=320\degree$*

*Angles in a quadrilateral add up to $360\degree$. To find the missing angle, subtract the angles from the total angle:*

$360-320=40\degree$

*Angle B = *$\underline{ 40\degree}$

## Angles in regular polygons

To work out the sum of all internal angles of a regular ploygon, you can use your knowledge of triangles. You can split all regular polygons into different triangles which do not intersect one another, from one vertex to another. As angles in a triangle add up to $180\degree$, the sum of all internal angles of a regular polygon is equal to $number\ of\ triangles\times 180$.

A formula can be used to make working this out easier. Take $n$ as the number of sides of the polygon. To work out the sum of all internal angles the formula is:

$sum\ of\ internal\ angles\ =(n-2)\times180$

This information can then be used to work out each internal angle in the polygon, by doing:

$Each\ internal\ angle = \dfrac{sum \ of \ internal \ angles}{n}$

##### Example 3

*Work out each internal angle of a regular octagon.*

*An octagon has eight sides: $n=8$.*

*The sum of the internal angles can be worked out by:*

*$(8-2)\times180$*

*$=6\times180=1080\degree$*

*Each internal angle:*

*$\dfrac{1080}{8}=135$*

*Each internal angle is $\underline{135\degree}$.*