Special quadrilaterals
A quadrilateral is any 4-sided polygon. Six special types appear constantly in GCSE — each with characteristic properties.
The six special quadrilaterals
Square
- Four equal sides.
- Four right angles.
- Diagonals equal, bisect each other at right angles, and bisect the angles.
Rectangle
- Opposite sides equal.
- Four right angles.
- Diagonals equal and bisect each other.
Rhombus
- Four equal sides.
- Opposite angles equal.
- Diagonals bisect each other at right angles, and bisect the angles.
Parallelogram
- Two pairs of parallel sides.
- Opposite sides equal; opposite angles equal.
- Diagonals bisect each other (NOT at right angles).
Trapezium
- One pair of parallel sides.
- An isosceles trapezium has the non-parallel sides equal — and one line of symmetry.
Kite
- Two pairs of adjacent equal sides.
- One pair of opposite angles equal.
- One diagonal bisects the other at right angles, and bisects two of the angles.
Sum of interior angles
For any quadrilateral, the four interior angles sum to 360°.
Symmetry summary
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Parallelogram | 0 | 2 |
| Isosceles trapezium | 1 | 1 |
| Kite | 1 | 1 |
✦Worked example
A parallelogram has angles x and 110°. Find x.
- Opposite angles equal; co-interior angles between parallel sides sum to 180°.
- x = 180 − 110 = 70°.
Identifying from clues
If a quadrilateral has 4 equal sides AND 4 right angles → square. If it has 4 right angles only → rectangle. If it has 4 equal sides only → rhombus. If diagonals bisect at 90° but only 2 pairs of equal sides → kite.
⚠Common mistakes
- Confusing rhombus with parallelogram — rhombus has equal sides; parallelogram doesn't necessarily.
- Saying parallelogram diagonals are equal — they aren't (only rectangles, squares, isosceles trapezia have equal diagonals).
- Treating trapezium and parallelogram as the same — trapezium has only ONE pair of parallel sides.
- Missing the diagonal-perpendicular property — it applies to square, rhombus, kite (NOT general parallelogram).
- Forgetting 360° — quadrilateral angles sum to 360°, not 180°.
➜Try this— Quick check
In a kite ABCD with AB = AD = 5 cm, BC = CD = 8 cm, and angle BAD = 70°. State the angle BCD if angle ABC = angle ADC.
- 360 − 70 − 2x = 0 (where x = ABC = ADC)... Actually: 70 + 2x + BCD = 360.
- We have one pair equal: ABC = ADC. So 70 + BCD + 2x = 360.
- One specific case: if x = 110°, BCD = 360 − 70 − 220 = 70°. (Special case — generally would need more info.)
AI-generated · claude-opus-4-7 · v3-deep-geometry