Year 11 Foundation · Angles, bearings and trigonometry
Work out angles in lines and shapes
Learn how to work out angles in shapes and on parallel lines, solve angle problems with algebra, scale similar shapes, handle bearings with drawings and right-angled triangle methods, and construct accurate right angles with compasses.
Definition
An angle measures an amount of turn between two lines, in degrees. On the Foundation paper you rarely measure an angle: you work it out from facts, such as angles on a straight line adding to 180 degrees, and those facts still hold when an angle is written with algebra or sits on a bearings diagram. A bearing describes a direction as a clockwise turn from north, and similar shapes keep all their angles the same while their lengths scale. When a question says construct, compasses and a ruler build exact right angles that a protractor sketch cannot match.
Why Do I Have to Learn This?
Angles measure turn, from door hinges to skate ramps. Reading them helps you build, draw and navigate accurately.
Key Rules:
- 1. Angles on a straight line add to 180 degrees, angles round a point add to 360 degrees, angles in a triangle add to 180 degrees, vertically opposite angles are equal, and where a line crosses parallel lines, alternate and corresponding angles are equal while co-interior angles add to 180 degrees.
Remember:
- •Bearings turn clockwise from north and take three figures; scale factors multiply, never add; and every angle you cannot measure comes from a fact you can quote.
Don’t Forget...
- •Measuring a bearing anticlockwise from north, or writing 70 degrees instead of 070. A bearing is defined as the clockwise turn from north and always carries three figures, so both slips name a different direction from the one intended.
- •Adding the same amount to every side of a similar shape instead of multiplying by the scale factor. Enlargement multiplies lengths: 7 grows to 17.5 by a factor of 2.5, so 6 becomes 6 x 2.5 = 15, not the 16.5 you get by adding 10.5.
- •Reaching for Pythagoras when the question gives or asks for an angle. Pythagoras only links the three sides of a right-angled triangle; the moment another angle is involved, sin, cos or tan is the rule that connects it to the sides.
Acute angle
An angle smaller than 90 degrees.
An angle of 45° is acute.
Worked Example:
The bearing of a ship B from a port A is 070 degrees. Work out the bearing of A from B.
Draw a north line at A and another at B; every north line points the same way, so the two are parallel.
The bearing of B from A is the clockwise angle at A between north and the line AB, which is 70 degrees.
Looking from B back to A faces the exact opposite way along the same line, which is half a turn more, so add 180 degrees.
70 + 180 = 250, and 250 already has three figures.
Answer: 250 degrees
Why It Works:
Every north line points due north, so any two of them are parallel and the parallel line rules apply between them. Reversing a direction is exactly half a turn, and half of 360 degrees is 180, so a return bearing always differs from the outward one by 180 degrees. The angle facts themselves come from turn: a straight line is half a full turn and angles round a point make a whole one.
Another Example:
Two triangles are similar. The smaller has sides 4 cm, 6 cm and 7 cm, and the longest side of the larger triangle is 17.5 cm. Work out the length of the side that matches the 6 cm side.
Match the sides first: the 17.5 cm side matches the 7 cm side, because each is the longest in its own triangle.
Divide to find the scale factor: 17.5 ÷ 7 = 2.5.
Multiply the matching side by the same factor: 6 x 2.5 = 15.
Answer: 15 cm
Animated Example:
Free Work out angles in lines and shapes Worksheet
Download free worksheets and practise today.
Every download has a fresh set of questions.
Before this: Angles (Year 10 Foundation)
Let’s Practise the Concept
Step 1 of 3
Put the angles in order
Tap them from smallest to largest.
In order so far: nothing yet