Angles
What is an Angle?
An angle is a measure of the amount of turn between two straight lines that meet at a point called the vertex. Angles are measured in degrees (°), where a full rotation equals 360°.
Angles appear everywhere in geometry — in shapes, parallel lines, bearings, and constructions. Understanding their properties and rules is essential for solving problems efficiently.
Types of Angles
| Type | Size | Description |
|---|---|---|
| Acute | 0° to 90° | Less than a right angle |
| Right | Exactly 90° | A perfect corner, marked with a small square |
| Obtuse | 90° to 180° | Greater than a right angle but less than a straight line |
| Straight | Exactly 180° | A flat line |
| Reflex | 180° to 360° | Greater than a straight angle |
| Full turn | Exactly 360° | A complete rotation |
When identifying angle types, always compare to the 90° benchmark first: less than 90° is acute, greater is obtuse (up to 180°), then reflex.
Key Angle Rules
These rules let you find missing angles without measuring:
Angles on a straight line — add up to 180°
- If one angle is 65°, the other on the same line is 180° - 65° = 115°
Angles around a point — add up to 360°
- Four angles at a point: if three are 80°, 100°, 70°, the fourth is 360° - 250° = 110°
Vertically opposite angles — when two lines cross, the angles directly opposite each other are equal
- If one angle is 42°, its vertically opposite angle is also 42°
Complementary angles — two angles that add up to 90°
Supplementary angles — two angles that add up to 180°
Angles in Triangles
The interior angles of any triangle always add up to 180°.
Worked example: A triangle has angles 47° and 63°. Find the third angle.
- 47 + 63 = 110
- Third angle = 180 - 110 = 70°
The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
- If interior angles are 40° and 70°, the exterior angle is 40 + 70 = 110°
Special triangles:
- Equilateral — all angles are 60°
- Isosceles — two equal base angles
- Right-angled — one angle is exactly 90°
Angles in Polygons
For any polygon, the sum of interior angles = (n - 2) × 180°, where n is the number of sides.
| Shape | Sides (n) | Interior angle sum |
|---|---|---|
| Triangle | 3 | 180° |
| Quadrilateral | 4 | 360° |
| Pentagon | 5 | 540° |
| Hexagon | 6 | 720° |
For a regular polygon (all sides and angles equal), each interior angle = sum ÷ n.
Angles and Parallel Lines
When a transversal (a line crossing two parallel lines) is drawn, four important angle relationships are created. In the diagrams below, lines p and q are parallel:
Alternate angles — on opposite sides of the transversal, between the parallel lines. They are equal.
- Also called "Z angles" because of the Z shape they form.
Corresponding angles — on the same side of the transversal, one between the parallels and one outside. They are equal.
- Also called "F angles".
Co-interior angles (same-side interior) — on the same side of the transversal, between the parallel lines. They add up to 180°.
- Also called "C angles" or allied angles.
Vertically opposite angles — formed at each intersection, equal at every crossing point.
Worked Example: Parallel Lines
Two parallel lines are cut by a transversal. One angle is 55°. Find the alternate, corresponding, and co-interior angles.
- Alternate angle = 55° (equal)
- Corresponding angle = 55° (equal)
- Co-interior angle = 180° - 55° = 125°
Bearings
A bearing is a direction measured clockwise from north, always written as a three-digit number.
- North = 000°
- East = 090°
- South = 180°
- West = 270°
Example: A ship travels on a bearing of 065°. To find the back-bearing (return direction), add 180°: 065° + 180° = 245°.
Bearings use the same angle rules — alternate, corresponding, and co-interior angles often appear in bearing problems involving parallel north lines.
Common Mistakes
- Confusing obtuse and reflex — obtuse is between 90° and 180°; reflex is between 180° and 360°.
- Misidentifying alternate and co-interior — alternate angles are equal; co-interior angles add to 180°.
- Forgetting the polygon formula — many students guess 360° for all polygons; it is only correct for quadrilaterals.
- Not writing reasons — in exam questions, always state the angle rule you used (e.g. "angles on a straight line").
- Bearing errors — forgetting to measure clockwise from north, or writing a two-digit bearing instead of three digits.
Tips and Tricks
- Draw and label diagrams clearly — marking equal angles with the same arc symbol prevents confusion.
- When finding multiple missing angles, work step by step and label each answer before moving on.
- For parallel line problems, extend or redraw the transversal line to make the Z, F, or C shape visible.
- Always check: if you have found all angles at a point, do they sum to 360°? If not, recheck your work.