Angles and constructions
A sum of 360°
Two completely different situations, one number. Angles round a point and angles in a quadrilateral both add to 360°.
Angles around a point
Turn all the way round and you have turned through 360°. So the angles that meet at a point always add up to 360°. If you know all but one of them, subtract from 360° to find the missing angle.
Angles in a quadrilateral
The four angles of ANY quadrilateral also add up to 360°. Three angles of a quadrilateral are each 85°. 3 × 85° = 255° The fourth angle is 360° − 255° = 105°
Using symmetry to help
A kite has a vertical line of symmetry, so the two side angles are equal. Given 135° at the top and 80° on one side: The other side angle is also 80°, by symmetry. 135° + 80° + 80° = 295° So the last angle is 360° − 295° = 65° Symmetry gives you an extra angle for free — always look for it.
When all the angles are equal
If all the angles round a point are the same size, divide 360° by how many there are. Three equal angles: 360° ÷ 3 = 120° each. Four equal angles: 360° ÷ 4 = 90° each.
Worked examples
Three angles at a point are 90°, 120° and 60°
- Angles at a point total 360°.
- 90 + 120 + 60 = 270.
- 360 − 270 = 90.
Answer: The fourth angle is 90°
Three angles of a quadrilateral are each 85°
- 3 × 85 = 255.
- A quadrilateral's angles total 360°.
- 360 − 255 = 105.
Answer: 105°
A kite with 135° at the top and 80° at one side
- Symmetry makes the other side angle 80° too.
- 135 + 80 + 80 = 295.
- 360 − 295 = 65.
Answer: 65°
Three equal angles at a point
- They must total 360°.
- 360 ÷ 3 = 120.
Answer: 120° each
Four equal angles at a point
- 360 ÷ 4 = 90.
Answer: 90° each — four right angles
Why symmetry is worth looking for
- A line of symmetry means two angles are equal.
- That is one angle you do not have to be told.
Answer: It gives you an extra angle for free
Where you see this in real life
- Slices of a pizza meet at the centre — however you cut it, the angles there total 360°.
- A tiled floor works because the angles meeting at each corner add to exactly 360°, with no gap and no overlap.
Key words
- Quadrilateral
- a shape with four straight sides
- Angles at a point
- angles that meet at the same point, totalling 360°
- Angle
- the amount of turn between two lines
- Degree
- the unit for measuring angles, written °
- Line of symmetry
- a fold line that makes both halves match