Introduction

Wherever lines meet, angles are created. Looking around, we can see angles everywhere. Sometimes angles give us valuable information about how we are seeing something, not just what we are seeing. For instance, consider the two pictures below.

From the angles of the door on the right, we know that we are not looking at the door face on, because they are not right angles as we know they should be.

In this unit you will learn more about measuring and recognizing types of angles.

Look at the

triangle

A triangle is a three-sided polygon.

triangle below. The vertices are labelled A, B, and C. The edges of the triangle are the line segments AB, BC, and AC.



The angles are also labelled as x, y, and z.



There is another way that we can refer to the angles. For example,

angle

An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .

angle y can also be written as . The symbol shows that we are referring to an angle, while the letter in the middle (A) shows at which

vertex

A vertex is defined as the common endpoint of two lines.

vertex or point the angle is situated. The first and last letters (B and C) indicate the starting points of the two lines between which the angle is measured. We can also write this angle as . The various ways of writing out the three angles in the triangle are:



The diagram below contains three line segments, eight named points and eight marked angles.

Angles are measured in degrees (symbol: °). When two lines meet (such as A and B in Fig.3), the size of the angle between them can be thought of as the amount of turning that is required to make one line lie on top of the other. Rotate line A about the vertex to see the number of degrees through which it has to be turned to lie on top of line B.




Click on the figure below to interact with the model.



There are 360 degrees in a full turn, or complete rotation – so, for example, a minute hand on a clock rotates through 360° every hour.


Skateboarders and snowboarders name some of their jumps by how many degrees they spin through. For example, a '180 jump' is one where they turn through 180° (a half turn). Therefore, they land facing backwards. Another example is the '720 jump' where they turn through 720° (), which is 2 full turns. Therefore, they land facing forwards.



One way to measure angles is by using a protractor. Notice that on the protractor below there are two scales – one running clockwise and the other running anticlockwise.



To measure the angle between two lines, place the centre of the protractor (where the cross is) over the point where the two lines meet (2).



Now, rotate the protractor so that the 0 marking lines up with one of the lines (3). Finally, read off the value crossed by the second line (OB), making sure to use the correct scale (4). This reading tells you the angle between the two lines.

Although we can use a protractor to measure angles exactly, sometimes it is useful to classify angles in a more general way. This can usually be done without measuring. Acute angles

Angles of less than a quarter turn or

right angle

An angle of 90°.

right angle (90°) are called acute angles. You can think of these as angles as being 'sharp'.

Obtuse angles

Angles of between a quarter and a half turn (90 to 180) are called obtuse angles. These are 'blunt' angles.

Reflex angles

Angles of between a half and a whole turn (180 to 360) are called reflex angles.

Look at the line pattern below.

We have seen that half a turn is 180. Another way of saying this is that there are 180 degrees on a

straight line

A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore one-dimensional.

straight line. Rotate the

line segment

A line segment is the set of points on the straight line between any two points, including the two endpoints themselves.

line segment in Fig.12 through a half turn to show this.


Click on the figure below to interact with the model.




As you can see, when the angle is at 180, there is a straight line between A and B.

So in the following example, the three angles shown must always add up to
180. Move points B and C to see this in action.


Click on the figure below to interact with the model.




Using facts like this often enables us to work out an angle's size or type without measuring it.

When two angles sum (add up) to 90, we call them complementary angles. For example, in the diagram below of a half-open door, the angles are complementary since they add up to 90 (because the walls are at 90 to each other).





When two angles sum to 180, we call them supplementary angles. For example, the angles either side of the base of a ladder are supplementary (assuming that the ground is level).


Look at the following pattern of lines.

Look at the two intersecting line segments below. Select the entire model then paste a copy of it (2). Now move it so that point C' joins to point B (3).



Do you notice anything special about the lines and angles?

You have produced a model of two

parallel

Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.

parallel line segments (CD and GH) intersected by a

transversal

A transversal line intersects two or more coplanar lines.

transversal (BE). By copying and pasting, you know that the

corresponding angles

Corresponding angles are created when a transversal or line segment intersects two other lines or line segments.

Angles a and e are corresponding angles. If the two intersected lines are parallel, then the corresponding angles are equal. They are also known as 'F' angles.

corresponding angles a and b are equal.

Whenever parallel lines (or line segments) are intersected by a transversal, the size of corresponding angles is equal.



When two lines are parallel, this is often shown by marking a small arrowhead on each line, as on the lines AB and CD in Fig.19 above.




Look at the line pattern below.



In the unit Types of Line it was shown that vertically

opposite angles

Opposite angles are formed when two lines intersect.

Angle a is opposite angle c and angle b is opposite angle d. Opposite angles are always equal.

opposite angles are always equal.



So we can see that . This means that the

alternate angles

Alternate angles are created when a transversal or line segment intersects two other lines or line segments.

Angles d and f are alternate angles. If the two intersected lines are parallel, then the alternate angles are equal. They are also known as 'Z' angles.

alternate angles a and c are of equal size.

In fact, this same pattern is created when any transversal crosses a pair of parallel lines. It is always true, therefore, that alternate angles located on parallel lines are of equal size.

We can use these facts about parallel lines to find missing angles such as below.


Follow the steps to find out how to calculate .

Summary

Angles are measured in degrees and can be acute, obtuse or reflex.

If two angles sum to 90, they are said to be complementary.

If two angles sum to 180, they are supplementary.

Corresponding angles on parallel lines are always equal. Alternate angles on parallel lines are also always equal.