Foundation Maths by Example

ANGLES

This angle is called ABC or AB̂C or ∠ABC.

A dash like this means the lines are of equal length.

If angles are marked like this, without the angle value given, it means they are the same size. Parallel lines are lines that never meet. They are marked with arrows. If there are two pairs

of parallel lines, the second pair are marked with a set of double arrows ( >> ).

Angles on a Straight Line

Angles on a straight line add up to 180°.

Find the value of x.

Example 1

Example 2

Example 3

Angles Around a Point

a + b + c + d = 360°

Angles around a point add up to 360°

Find the missing angles.

Example 1

Example 2

Example 3

Practice Questions

Use the previous examples to help you with these questions.

For each of the following, find the missing angles.

Angles in a Triangle

Finding Angles in Triangles

a + b + c = 180°

‘Angles in a triangle add up to 180°’

Example 1

Example 2

The dashes means it’s an isosceles triangle

So y is 50°

Example 3

We know this angle because ‘angles on a straight line add up to 180°’

180° − 110° = 70°

Vertically opposite angles

‘When two straight lines cross, the opposite angles are equal.’

Example 4

The dashes means it is an isosceles triangle.

b is vertically opposite the 40° angles, so b is also 40°

let a = c = x

Practice Questions

Find the missing angles in each question.

Angles in Parallel Lines

Parallel lines are lines that never meet, they remain the same distance apart. They are marked with ( > ), to show they are parallel.

The line that crosses the parallel lines is called the transversal.

Parallel lines often have an acute angle and an obtuse angle.

Learn these:

This is not a valid reason in your exam, you must use the keywords.

Some students remember

Corresponding

Interior

Alternate

F–angle

C–angle or U–angle

Z–angle

Example 1

Example 2

Example 3

Multistep Examples

1

2

3

Practice Questions

For each question find the value of the missing angle and state the reason.

Scale Drawings and Bearings

Bearings

Bearings are often used in navigation to describe a direction or a change of direction.

Rules

1. Always three figures (eg 120 or 0

70).

2. Measured in a clockwise direction.

3. Measured from north.

Compass

Bearings and navigation use a compass. You need to know the points of the compass and be able to work out the three−figure bearing of them.

Measuring a Bearing

You will need a protractor – when measuring a bearing, it should be orientated so that the zero is facing to the north (normally up the page).

Example 1: Acute or Obtuse

Measure the bearing of B from A

‘from A’ means put the protractor on A

Start from zero and go clockwise

Must be three figures

Example 2: Reflex angle

Method

1. Measure angle labelled x

2. Subtract x from 360°

360° − x = bearing of B from A

1. x = 110°

2. 360° − 110° = 250°

Answer = 250°

Practice Questions

For each question measure the bearing of B from A.

Calculating a Return Bearing

A return bearing is the bearing in which you would have to travel to get back to the start.

Misconception

A lot of students think you should subtract the bearing you have been given from 360°.

Example 1

What is the bearing of A from B?

Why? Earlier in this chapter we did parallel lines and it is a Z–angle so it is the same as the angle at A.

Example 2

What is the bearing of A from B?

Remember

If angle (bearing) is less than 180° then you need to add 180° to find the return bearing. If angle is more than 180°, you subtract 180° to find the return bearing.

Practice Questions

For each of the following questions find the return bearing (a from b).

Scale and Maps

A scale is a way of normally shrinking down the size of an object or distance, so that it can be drawn on a piece of paper.

For example:

On a map you often get a scale of 1:25,000. This means 1 cm on the map is equal to 25,000 cm in real life – 250 m.

Remember

1 cm = 10 mm

1 m = 100 cm

1 km = 1000 m

Example 1

Scale 1 : 25,000

If two towns are 8 cm apart on the map, how far are they apart in real life? 1 : 25,000

8                                        × 8

Distance on map          In the ratio, do the same to both sides

8 : ?

So 25,000 × 8 = 200,000 cm

Converting to km

200,000 cm → 2,000 m → 2 km

÷ 100 ÷ 1,000

Example 2

Scale 1 : 50,000

If two towns are 6 cm apart on a map, how far are they apart in real life?

1 : 50,000

× 6               × 6

6 : 300,000

300,000 cm → 3,000 m → 3 km

Example 3

Scale 1 : 30,000

If two ships are 15 km apart in real life on the sea, how far would this distance appear on a map (or sometimes called a chart)?

Convert 15 km into cm

15 km → 15,000 m → 1500,000 cm

      × 1,000       × 100

Scale

1 : 30,000

? : 1500,000

Multiply by same number

Divide

So 1 × 50 = 50 cm on the map

Example 4

Scale 1 cm : 2 km

A ship is 8 km from a buoy on a bearing of 070°

Mark the position on the map

Method

1. Draw north line from the buoy

2. Place protractor on north line

3. Measure 70° from north, clockwise

4. Draw line at correct angle

5. Scale is 1cm : 2km so 8km ÷ 2km is represented by 4cm. Measure 4 cm along it and mark the location of the ship.