Mathematics 0580 · Syllabus tracker
Polygons
Angles on lines and at points, angles with parallel lines, angle sums of triangles, quadrilaterals and polygons, and giving reasons.
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Quick revision
Polygons on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Angle facts, parallel lines, angles in polygons, and how to write geometric reasons.
1Basic angle facts
- Angles on a straight line add to 180°. Angles at a point add to 360°.
- Vertically opposite angles are equal.
- Angles in a triangle add to 180°; in a quadrilateral, 360°.
- Isosceles triangle: two equal sides and the two base angles equal. Equilateral: all angles 60°.
- The exterior angle of a triangle equals the sum of the two opposite interior angles.
Worked example
Three angles on a straight line are 2x, 3x and 4x. Find x.
- 2x + 3x + 4x = 180.
- 9x = 180.
x = 20
2Angles and parallel lines
- Alternate angles (Z-shape) are equal.
- Corresponding angles (F-shape) are equal.
- Co-interior angles (C-shape) add to 180°.
- Write the reason in words: 'alternate angles are equal' — not 'Z angles'.
| Pair | Shape | Rule |
|---|---|---|
| Alternate | Z | equal |
| Corresponding | F | equal |
| Co-interior | C / U | add to 180° |
| Vertically opposite | X | equal |
Worked example
Two parallel lines are crossed by a transversal. One co-interior angle is 115°. Find the other.
- Co-interior angles add to 180°.
- 180 − 115.
65°
3Angles in polygons
- Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
- Exterior angles of any polygon add to 360°.
- Regular polygon: exterior angle = 360° ÷ n; interior = 180° − exterior.
- Number of sides of a regular polygon = 360° ÷ exterior angle.
| Polygon | Sides | Angle sum | Regular interior |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Formulas
Worked example
The interior angle of a regular polygon is 5 times its exterior angle. How many sides does it have?
- Interior + exterior = 180, so 6 × exterior = 180.
- Exterior = 30°.
- n = 360 ÷ 30.
12 sides
4Giving reasons
- Use three-letter notation: angle ABC is the angle at B.
- Each step: the angle, then its reason. For example: 'angle ABD = 40° (alternate angles are equal)'.
- Common reasons: angles on a straight line, angles in a triangle, isosceles triangle, alternate / corresponding / co-interior angles, angles in a polygon.
Worked example
In triangle ABC, AB = AC and angle BAC = 40°. Find angle ABC, giving a reason.
- Base angles of an isosceles triangle are equal.
- (180 − 40) ÷ 2 = 70.
70° (base angles of an isosceles triangle are equal; angles in a triangle add to 180°)
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Regular hexagon interior angle = 360 ÷ 6 = 60° | That's the exterior angle; interior = 120° |
| Using n × 180 for the angle sum | (n − 2) × 180 |
| Writing 'Z angles' as a reason | Write 'alternate angles are equal' |
| Saying co-interior angles are equal | They add to 180° |
★In the exam
- Mark angles on the diagram as you find them.
- 'Give reasons' questions lose marks without the words — write every reason.
- Diagrams are not drawn to scale: never measure; calculate.