# Points, lines and angles  (Page 2/3)

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Angles are measured in degrees which is denoted by ${}^{\circ }$ , a small circle raised above the text in the same fashion as an exponent (or a superscript).

Angles can also be measured in radians. At high school level you will only use degrees, but if you decide to take maths at university you will learn about radians.

## Measuring angles

The size of an angle does not depend on the length of the lines that are joined to make up the angle, but depends only on how both the lines are placed as can be seen in [link] . This means that the idea of length cannot be used to measure angles. An angle is a rotation around the vertex.

## Using a protractor

A protractor is a simple tool that is used to measure angles. A picture of a protractor is shown in [link] .

Method:

Using a protractor

1. Place the bottom line of the protractor along one line of the angle so that the other line of the angle points at the degree markings.
2. Move the protractor along the line so that the centre point on the protractor is at the vertex of the two lines that make up the angle.
3. Follow the second line until it meets the marking on the protractor and read off the angle. Make sure you start measuring at 0 ${}^{\circ }$ .

## Special angles

What is the smallest angle that can be drawn? The figure below shows two lines ( $CA$ and $AB$ ) making an angle at a common vertex $A$ . If line $CA$ is rotated around the common vertex $A$ , down towards line $AB$ , then the smallest angle that can be drawn occurs when the two lines are pointing in the same direction. This gives an angle of 0 ${}^{\circ }$ . This is shown in [link]

If line $CA$ is now swung upwards, any other angle can be obtained. If line $CA$ and line $AB$ point in opposite directions (the third case in [link] ) then this forms an angle of 180 ${}^{\circ }$ .

If three points $A$ , $B$ and $C$ lie on a straight line, then the angle between them is 180 ${}^{\circ }$ . Conversely, if the angle between three points is 180 ${}^{\circ }$ , then the points lie on a straight line.

An angle of 90 ${}^{\circ }$ is called a right angle . A right angle is half the size of the angle made by a straight line (180 ${}^{\circ }$ ). We say $CA$ is perpendicular to $AB$ or $CA\perp AB$ . An angle twice the size of a straight line is 360 ${}^{\circ }$ . An angle measuring 360 ${}^{\circ }$ looks identical to an angle of 0 ${}^{\circ }$ , except for the labelling. We call this a revolution .

## Angles larger than 360 ${}^{\circ }$

All angles larger than 360 ${}^{\circ }$ also look like we have seen them before. If you are given an angle that is larger than 360 ${}^{\circ }$ , continue subtracting 360 ${}^{\circ }$ from the angle, until you get an answer that is between 0 ${}^{\circ }$ and 360 ${}^{\circ }$ . Angles that measure more than 360 ${}^{\circ }$ are largely for mathematical convenience.

• Acute angle : An angle $\ge {0}^{\circ }$ and $<{90}^{\circ }$ .
• Right angle : An angle measuring ${90}^{\circ }$ .
• Obtuse angle : An angle $>{90}^{\circ }$ and $<{180}^{\circ }$ .
• Straight angle : An angle measuring 180 ${}^{\circ }$ .
• Reflex angle : An angle $>{180}^{\circ }$ and $<{360}^{\circ }$ .
• Revolution : An angle measuring ${360}^{\circ }$ .

These are simply labels for angles in particular ranges, shown in [link] .

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