Properties of special quadrilaterals
OCR J560 explicitly tests recall of quadrilateral properties. Foundation questions ask for matching properties to names; Higher uses these properties to set up algebraic equations and proofs.
The hierarchy
A quadrilateral is any 4-sided polygon. Special types form a hierarchy:
- Trapezium — exactly one pair of parallel sides.
- Isosceles trapezium — non-parallel sides equal in length; line of symmetry.
- Parallelogram — both pairs of opposite sides parallel.
- Rhombus — parallelogram with all 4 sides equal.
- Rectangle — parallelogram with all 4 angles 90°.
- Square — both rhombus AND rectangle (all sides equal AND all angles 90°).
- Kite — two pairs of adjacent equal sides.
Property table
| Shape | Sides | Angles | Diagonals | Lines of sym | Order rot sym |
|---|---|---|---|---|---|
| Square | All 4 equal | All 90° | Equal, perpendicular, bisect | 4 | 4 |
| Rectangle | Opposite equal | All 90° | Equal, bisect each other | 2 | 2 |
| Rhombus | All 4 equal | Opposite equal | Perpendicular, bisect each other | 2 | 2 |
| Parallelogram | Opposite equal | Opposite equal | Bisect each other | 0 | 2 |
| Kite | Two pairs adjacent equal | One pair equal | Perpendicular | 1 | 1 |
| Trapezium | One pair parallel | — | — | 0 (or 1 for iso) | 1 |
| Iso. trapezium | Non-parallel sides equal | Base angles equal | Equal | 1 | 1 |
Angle facts
Sum of interior angles of any quadrilateral = 360°.
Diagonals: parallelograms' diagonals always bisect each other but don't always have equal length (only rectangle and square).
Algebra applications
OCR Higher often gives a quadrilateral with side lengths in algebraic form and asks you to find the unknown.
Example: a rectangle has sides 3x − 1 and x + 5. The opposite sides must be equal, but adjacent sides need not be — so the question must specify what's given. Often the perimeter is given; set 2(3x − 1) + 2(x + 5) = perimeter and solve.
OCR mark scheme conventions
- B1 for correct identification of a quadrilateral by properties.
- M1 + A1 for setting up and solving an equation using property + given.
- "Justify" requires the property to be NAMED (e.g. "opposite sides of parallelogram are equal").
⚠Common mistakes
- Confusing rhombus and rectangle (both are parallelograms but distinguish all-sides-equal vs all-angles-90°).
- Forgetting that a square is also a rectangle (and a rhombus).
- Stating "diagonals are equal" for a parallelogram (only true for rectangles and squares).
- Saying "trapezium" when meaning "isosceles trapezium" — properties differ.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves