Module 3
Angles, polygons & constructions
Quick-reference revision notes for parents.
3.0 Angles in a quadrilateral
Any four-sided shape: angles add to 360°.
A quadrilateral has angles 80°, 110°, 95° and x. Find x.
x = 360 − 80 − 110 − 95 = 75°
3.1 Parallel lines
When a straight line (transversal) crosses two parallel lines, three angle relationships appear:
(equal)
(equal)
(a + b = 180°)
F = corresponding (equal). Z = alternate (equal). C = co-interior (add to 180°).
3.2 Polygons
Sum of interior angles of a polygon with n sides:
(n − 2) × 180°
Sum of exterior angles of any polygon = 360°.
For a regular polygon (all sides and angles equal):
- Each exterior angle = 360 ÷ n
- Each interior angle = 180 − exterior angle
Multiplying both sides by n gives a handy shortcut:
(exterior angle) × (number of sides) = 360°
Especially useful when you're given an exterior angle and asked to find n: just divide 360 by the angle.
Sum interior = (6 − 2) × 180 = 720°
Each interior = 720 ÷ 6 = 120°
Each exterior = 360 ÷ 6 = 60°
3.3 Using a protractor
- Place the centre cross on the angle's vertex
- Line up 0° with one of the arms
- Read where the other arm crosses — use the scale that starts at 0° on that arm
Two scales on the protractor (inner and outer). Always start from 0° — the wrong scale gives an angle that's 180 − x.
3.4 Bearings
A bearing is a direction measured clockwise from North, written as three digits.
North = 000° (or 360°)
East = 090°
South = 180°
West = 270°
North-east = 045°
(clockwise from N).
Back bearings
The bearing of A from B is called the back bearing. The two North lines are parallel, so the angles on either side of line AB are co-interior — they add to 180°.
To find the back bearing (bearing of A from B, measured clockwise from B's North):
back bearing = 360° − 120° = 240°
Once you spot that interior angles add to 180°: back bearing = forward bearing ± 180° (add 180° if the forward bearing is less than 180°, subtract otherwise).
3.5 Scale drawings
A scale of 1 : 100 means 1cm on the drawing represents 100cm in real life.
Map scale 1 : 50 000. A road is 6cm long on the map. Real length?
6 × 50 000 = 300 000 cm = 3 km
3.6 Constructing triangles
Three common cases:
| Given | Tools |
|---|---|
| Three sides (SSS) | Ruler + compasses |
| Two sides + included angle (SAS) | Ruler + protractor |
| One side + two angles (ASA) | Ruler + protractor |
Special triangles
Equilateral: all sides equal → all angles equal → each is 60°.
Isosceles: equal sides face equal angles. If you know the apex angle a, each base angle is (180 − a) ÷ 2.
Right-angled: contains a 90° angle. The longest side (the hypotenuse) is opposite it.
Scalene: nothing equal — but all 3 angles still sum to 180°.
Quick reference
- Quadrilateral angles sum to 360°
- Triangle angles sum to 180°
- Polygon: interior sum = (n − 2) × 180°
- Exterior angles always sum to 360°
- F (corresponding) and Z (alternate) angles are equal; C (co-interior) sum to 180°
- Bearings: clockwise from North, three digits