Angles: Parallel Lines, Polygons & Bearings
Angles: Parallel Lines, Polygons & Bearings
Angle Facts
- Angles on a straight line add up to 180°
- Angles around a point add up to 360°
- Vertically opposite angles are equal
- Angles in a triangle add up to 180°
- Angles in a quadrilateral add up to 360°
Parallel Lines
When a line (called a transversal) crosses two parallel lines, it creates several angle relationships:
- Alternate angles (Z-angles): equal, on opposite sides of the transversal, between the parallel lines
- Corresponding angles (F-angles): equal, on the same side, one between and one outside the parallel lines
- Co-interior angles (C-angles or allied angles): add up to 180°, on the same side, between the parallel lines
Worked Example: A transversal crosses two parallel lines. One angle is 65°. Find the alternate angle and the co-interior angle.
- Alternate angle = 65° (alternate angles are equal)
- Co-interior angle = 180° − 65° = 115° (co-interior angles sum to 180°)
Angles in Polygons
Interior angles are inside the polygon. Exterior angles are formed by extending one side.
Key formulas:
- Sum of interior angles = (n − 2) × 180° where n = number of sides
- Sum of exterior angles = 360° (always, for any polygon)
- Each interior + its exterior = 180°
For a regular polygon (all sides and angles equal):
- Each interior angle = (n − 2) × 180° ÷ n
- Each exterior angle = 360° ÷ n
| Polygon | Sides | Interior Angle Sum | Each Interior (regular) |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Worked Example: The interior angle of a regular polygon is 156°. How many sides does it have?
- Exterior angle = 180° − 156° = 24°
- Number of sides = 360° ÷ 24° = 15 sides
Worked Example: Find angle x in a pentagon where the other four angles are 100°, 130°, 95° and 110°.
- Sum of interior angles = (5 − 2) × 180° = 540°
- x = 540° − 100° − 130° − 95° − 110° = 105°
Bearings
A bearing is a direction measured clockwise from North, always written as a three-figure number.
Rules for bearings:
- Always measure from North
- Always measure clockwise
- Always give three figures (e.g. 045° not 45°)
Common bearings: North = 000°, East = 090°, South = 180°, West = 270°.
Worked Example: The bearing of B from A is 130°. Find the bearing of A from B.
- The bearing of A from B is the "return bearing"
- 130° + 180° = 310°
Rule: If the bearing from A to B is less than 180°, add 180°. If it is more than 180°, subtract 180°.
Worked Example: A ship sails on a bearing of 065° for 80 km, then on a bearing of 155° for 60 km. Find the bearing and distance from the starting point.
This requires a scale drawing or trigonometry:
- Draw North at the starting point, measure 065° clockwise, draw 80 km to scale
- From that point, draw North again, measure 155° clockwise, draw 60 km
- Measure back to start for distance and bearing
Exam Tips
- Always give reasons for angle calculations — write "alternate angles", "angles in a triangle" etc.
- For bearings, draw North lines at EVERY point
- The exterior angle shortcut (360° ÷ n) is the fastest way to find the number of sides of a regular polygon
- Bearings must always be three figures — write 070° not 70°