Working with Angles
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.

Naming 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.

Which two line segments meet at
point
• A point has no properties except position. It is an object with zero dimensions.
• Points in the x-y plane can be specified using x and y coordinates.
point
B?
 Figure 1. Naming vertices and angles.

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

Match each angle to the vertex where it is located.
•  Vertex A angle x angle y angle z Vertex B angle x angle y angle z Vertex C angle x angle y angle 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.

 Figure 2. Lines, points, and angles.
Match each angle with the correct letter from Fig.2.
•  angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h angle a angle b angle c angle d angle e angle f angle g angle h
Measuring 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.

 Figure 3.  Angle measurement.

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.
Since there are 360 degrees in a full turn, write down how many degrees are in these turns:

• A turn is .

A turn is .

A turn is .

A turn is .

Two turns is .

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.

 Figure 4. A snowboarder in the middle of a jump.
For each of these jumps, decide whether the snowboarder will land facing forwards or backwards.
•  A 360 jump backwards forwards A 720 jump backwards forwards A 180 jump backwards forwards A 540 jump backwards forwards A 900 jump backwards 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.

 Figure 5. A protractor.

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).

 Figure 6. Using a protractor.

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.
What is the angle between the two lines on Fig.6?
 Figure 7. Measuring angles.

Use the protractor to measure the values of the various angles marked in Fig.7 above.
•  Angle a 165 degrees 35 degrees 50 degrees 60 degrees Angle b 165 degrees 35 degrees 50 degrees 60 degrees Angle c 165 degrees 35 degrees 50 degrees 60 degrees Angle d 165 degrees 35 degrees 50 degrees 60 degrees

Types of angle
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
 Figure 8. Examples of 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
 Figure 9. Examples of obtuse angles.

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

Reflex angles
 Figure 10. Examples of reflex angles.

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

Look at the line pattern below.

 Figure 11. A line pattern.
Match the angle to its type.
•  Angle a acute obtuse reflex Angle b acute obtuse reflex Angle c acute obtuse reflex Angle d acute obtuse reflex Angle e acute obtuse reflex Angle f acute obtuse reflex Angle g acute obtuse reflex Angle h acute obtuse reflex
Angles on a straight line
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.

 Figure 12.  A straight line as a 180° angle.

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.

 Figure 13.  Angles on a straight line sum to 180°.

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

 Figure 14. Work out the value of each lettered angle.

Work out the missing values for the angles in Fig.14 above.

• Angle a is .

Angle b is .

Angle c is .
Complementary and supplementary angles
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).

 Figure 15. A door making complementary angles.
Angles a and b are complementary. When angle a is equal to 67 what is the value of angle b?

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).
 Figure 16. A ladder illustrating supplementary angles.

Look at the following pattern of lines.

 Figure 17. A line pattern.
Choose the correct angles or words to complete the sentence.

• Angle is complementary to b while is supplementary to it. Angles b and c sum to . Angle c is d's and b is its . Angles a and d sum to .
Angles on parallel 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).

 Figure 18. Creating corresponding angles.

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.

 Figure 19. Parallel lines AB and CD.

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.

 Figure 20. Corresponding angles of parallel lines.
Look at Fig.20 and pair each of the angles below with its value.
•  Angle a 107° 130° 50° 73° Angle b 107° 130° 50° 73° Angle c 107° 130° 50° 73° Angle d 107° 130° 50° 73°

Look at the line pattern below.

 Figure 21. Two parallel line segments cut by a transversal.

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.

Complete the following sentence by entering the correct angle labels.

• Using our knowledge of corresponding angles on parallel lines, we know that angle a is equal to angle . Using our knowledge of vertically opposite angles, we also know that angle b is equal to angle . Therefore a = = c.

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.
 Figure 22. Two parallel line segments cut by a transversal.

Follow the steps to find out how to calculate .
 First we use the fact that there are 180 on a straight line to calculate angle
 Then we use the fact that alternate angles are equal to determine the value we are looking for.

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.
Exercises
1. Fill in the spaces in the table below with the missing angles and angle names marked in Fig.23.

•  Angle Angle name r
 Angle Angle name x s

 Figure 23. Naming angles.
2. How many turns are equal to each of these angles?
•  270 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 45 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 3600 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 180 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 720 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 540 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 360 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns 90 an eighth of a turn half a turn one and a half turns one quarter of a turn one turn ten turns three quarters of a turn two turns

 Figure 24. Using a protractor to measure angles.
3. Use the protractor in Fig.24 to measure the angles there, then fill in the angle sizes in the table below.

•  Angle name Angle size (in degrees) Angle a Angle b Angle c Angle d Angle e Angle f Angle g Angle h

 Figure 25. Acute, obtuse, and reflex angles.

4. Decide whether each marked angle in Fig.25 above is
acute
An angle is acute if it is less than 90.
acute
,
obtuse
An angle is obtuse if it is over 90 but less than 180.
obtuse
or
reflex
A reflex angle is over 180 but less than 360.
reflex
.
•  Angle q acute obtuse reflex Angle r acute obtuse reflex Angle s acute obtuse reflex Angle t acute obtuse reflex Angle u acute obtuse reflex Angle v acute obtuse reflex Angle w acute obtuse reflex Angle x acute obtuse reflex Angle y acute obtuse reflex Angle z acute obtuse reflex
5. Use the fact that there are 180 on a straight line and 360 in a full turn to find the angles marked with small letters in the diagrams below. (The line PQ, where present, is always a straight line.)

• angle a =
angle b =
angle c =
angle d =
angle e =
angle f =
angle g =
angle h =
angle i =
angle j =

 Figure 26. Complementary and supplementary angles. Lines PQ and RS are straight lines.
6. Select the correct angles in Fig.26 to complete the sentences below.

• Angles c and d are angles. The pair of angles b and c are . Two angles in the figure are angles a and b.

7. Use what you know about opposite, corresponding, and alternate angles along with your knowledge of angles on a straight line to find the angles marked with small letters in the diagrams below. (The lines PQ and RS, where present, are always straight lines.)

• angle q =
angle r =
angle s =
angle t =
angle u =
angle v =
angle w =
angle x =
angle y =
angle z =

Well done!
Try again!