Properties of Polygons
Properties of Polygons
Special quadrilaterals and triangles each have a defined set of properties relating to their sides, angles, diagonals and symmetry. Knowing these properties — and which are sufficient to identify a shape — is essential for geometric reasoning.
Special Quadrilaterals
| Shape | Sides | Angles | Diagonals | Symmetry |
|---|---|---|---|---|
| Square | 4 equal, all parallel pairs | All 90° | Equal, bisect at 90° | 4 lines, order 4 |
| Rectangle | Opp. sides equal and parallel | All 90° | Equal, bisect each other | 2 lines, order 2 |
| Rhombus | 4 equal, opp. sides parallel | Opp. equal; adj. supp. | Bisect at 90° (unequal) | 2 lines, order 2 |
| Parallelogram | Opp. sides equal and parallel | Opp. equal; adj. supp. | Bisect each other (unequal) | None, order 2 |
| Trapezium | One pair of parallel sides | Co-interior pairs sum 180° | — | None (0 or 1 if isosceles) |
| Kite | Two pairs of adjacent equal sides | One pair of opp. equal angles | One bisects the other at 90° | 1 line, order 1 |
Triangle Types
| Type | Sides | Angles |
|---|---|---|
| Equilateral | 3 equal | All 60° |
| Isosceles | 2 equal | 2 base angles equal |
| Scalene | All different | All different |
| Right-angled | Any combination | One angle exactly 90° |
Polygon Names
Triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
Worked Examples
A quadrilateral has all four sides equal but angles are not 90°. Name it and state its diagonal properties.
Rhombus. Diagonals bisect each other at 90° but are not equal in length. Each diagonal bisects a pair of opposite angles.
A quadrilateral has one pair of parallel sides and non-parallel sides that are equal. Name it and state its angle properties.
Isosceles trapezium. Base angles are equal. Co-interior angle pairs (between the parallel sides) each sum to 180°.
In parallelogram \( ABCD \), \( \angle A = 67° \). Find all other angles, giving reasons.
\( \angle C = 67° \) (opposite angles in a parallelogram are equal).
\( \angle B = \angle D = 180 - 67 = 113° \) (co-interior angles, \( AD \parallel BC \); or adjacent angles in a parallelogram are supplementary).
Key Takeaways
- A square is a special rectangle, rhombus and parallelogram — it satisfies all their properties.
- In a parallelogram: opposite sides and angles are equal; adjacent angles are supplementary; diagonals bisect each other.
- A rhombus has all properties of a parallelogram plus all sides equal and diagonals meeting at 90°.
- A kite has one diagonal that is a line of symmetry, bisecting the other diagonal at 90°.