Angles: types, angle sum, angles on a line
We work out an unknown angle from a sum we know, and subtract the known angles. The angles in a triangle add up to 180°, in a quadrilateral 360°. Angles on a straight line add up to 180°. Vertically opposite angles are equal.
Watch Fox work it out on the board
Fox works one step at a time. Press play.
Types of angle
Angles are measured in degrees (°). A full turn is 360°, a straight line 180°.
| Type of angle | Size | Example |
|---|---|---|
| Acute angle | Less than 90° | 35°, 80° |
| Right angle | Exactly 90° | The corner of a sheet of paper |
| Obtuse angle | Between 90° and 180° | 120°, 150° |
| Straight angle | Exactly 180° | A straight line |
| Full turn | 360° | A clock hand goes all the way round |
Angle sum of triangles and quadrilaterals
Triangle: v₁ + v₂ + v₃ = 180°
Tear off the three corners of a triangle and put them next to each other. They make a straight line, 180°.
Quadrilateral: v₁ + v₂ + v₃ + v₄ = 360°
A diagonal splits the quadrilateral into two triangles: 180° + 180° = 360°.
- Equilateral triangle: all the angles are 180° ÷ 3 = 60°.
- Isosceles triangle: the two base angles are equal.
- Rectangle: all four angles are 90°.
Angles on a line and vertically opposite angles
- Angles on a straight line: next to each other, sum 180°.
- Vertically opposite angles: opposite each other where two lines cross, equal.
- All four angles around the crossing point: sum 360°.
Vertically opposite angles are equal because both lie on a straight line with the same angle. If that angle is 70°, both are 180° − 70° = 110°.
Example
Example 1: the third angle in a triangle
A triangle has angles of 48° and 75°. We want the third angle v on its own.
The three angles add up to 180°.
48° + 75° + v = 180°
We add the known angles.
123° + v = 180°
We subtract 123° from both sides, so the two sides stay equal.
123° − 123° + v = 180° − 123°
We work out the difference.
v = 57°
48° + 75° + 57° = 180°. That is correct.
Example 2: two lines that cross
Two straight lines cross and one angle is 64°. We want to find the other three.
The angle next to 64° lies on a straight line with it. Together they are 180°.
180° − 64° = 116°
The angle opposite 64° is vertically opposite and equal.
64°
The last one is opposite 116°, so it is also 116°.
116°
Around the crossing point: 64°, 116°, 64° and 116°.
A full turn: 64° + 116° + 64° + 116° = 360°. That is correct.
Pitfalls
Using 360° as the angle sum of a triangle, or 180° for a quadrilateral.
Count the corners. Three corners: 180°. Four corners are two triangles: 360°.
Answering 116° for the angle opposite 64°.
Opposite (vertically opposite angle) is equal: 64°. Next to it on a straight line gives 116°.
Reading the wrong scale on the protractor, so that 40° becomes 140°.
If the angle is smaller than the corner of a sheet of paper, the answer should be below 90°. Read the scale that starts at 0 on one of the lines.
Practise on your own
0 of 4 correct1An angle is 38°. How big is the angle next to it on a straight line?
2A right-angled triangle has one angle that is 27°. How big is the third angle?
3An isosceles triangle has an apex angle of 40°, the angle between the two equal sides. How big is each of the base angles?
4A quadrilateral has angles of 72°, 108° and 72°. How big is the fourth angle?
Common questions
What is the angle sum of a pentagon?
A pentagon splits into three triangles: 3 · 180° = 540°. In general, the sum is (number of corners − 2) · 180°.
Can a triangle have two right angles?
No. 90° + 90° = 180°, so nothing is left for the third angle.
What is the difference between angles on a straight line and vertically opposite angles?
Angles on a straight line are next to each other and add up to 180°. Vertically opposite angles are opposite each other and are equal.
Read also
- Years 8–9 and upper secondaryPythagoras' theorem step by stepWhat does Pythagoras' theorem say and how do you use it?
- Years 4–9Area and perimeterHow do you work out area and perimeter?
- Years 6–9Solving equations step by stepHow do you solve an equation?
- Years 5–9Scale: maps, drawings and enlargementsHow do you calculate with scale?
Keep practising with Somokai
Free to start. Hints and explanations when you get stuck.
Start practising for freeFound a mistake? Get in touch