Question
Draw the family tree of quadrilaterals. How are parallelograms, rectangles, rhombuses, squares, and trapeziums related? What properties does each type have?
(CBSE Classes 8-9 — quadrilateral properties tested for 5-8 marks)
Solution — Step by Step
Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a quadrilateral. The hierarchy from general to specific:
Quadrilateral → Trapezium → Parallelogram → Rectangle / Rhombus → Square
A square is BOTH a rectangle AND a rhombus. It sits at the top of the specialisation chain.
A quadrilateral with both pairs of opposite sides parallel.
- Opposite sides are equal
- Opposite angles are equal
- Diagonals bisect each other
- Adjacent angles are supplementary (sum = 180°)
Rectangle = Parallelogram + all angles 90°. Extra property: diagonals are equal.
Rhombus = Parallelogram + all sides equal. Extra property: diagonals bisect each other at 90°.
Square = Rectangle + Rhombus = all sides equal + all angles 90°. Has ALL properties of both.
Trapezium: Only one pair of opposite sides is parallel. The parallel sides are called bases. An isosceles trapezium has equal non-parallel sides.
Kite: Two pairs of adjacent sides are equal (not opposite sides). Diagonals are perpendicular, but only one diagonal is bisected by the other.
flowchart TD
A["Quadrilateral<br/>(4 sides)"] --> B["Trapezium<br/>(1 pair parallel sides)"]
A --> C["Kite<br/>(2 pairs adjacent equal)"]
B --> D["Parallelogram<br/>(2 pairs parallel sides)"]
D --> E["Rectangle<br/>(all angles 90°)"]
D --> F["Rhombus<br/>(all sides equal)"]
E --> G["Square<br/>(all sides equal + all angles 90°)"]
F --> GWhy This Works
The family tree shows that each special quadrilateral is a subset of a more general one. A square has ALL the properties of a parallelogram, a rectangle, AND a rhombus — because it satisfies all their defining conditions.
This hierarchy is useful for proving theorems: any property proved for parallelograms automatically applies to rectangles, rhombuses, and squares. You do not need to prove it separately for each.
Alternative Method
| Property | Parallelogram | Rectangle | Rhombus | Square |
|---|---|---|---|---|
| Opposite sides parallel | Yes | Yes | Yes | Yes |
| Opposite sides equal | Yes | Yes | Yes | Yes |
| All sides equal | No | No | Yes | Yes |
| All angles 90° | No | Yes | No | Yes |
| Diagonals bisect each other | Yes | Yes | Yes | Yes |
| Diagonals equal | No | Yes | No | Yes |
| Diagonals perpendicular | No | No | Yes | Yes |
A square has “Yes” for every property in the table above. This is the quickest way to remember: start with the square and remove one condition to get a rectangle or rhombus.
Common Mistake
Students say “a square is not a rectangle.” This is wrong. Every square IS a rectangle (it has all right angles and equal opposite sides). It is a special case of a rectangle where all four sides are also equal. Similarly, every square is a rhombus, every rectangle is a parallelogram, and so on. The family tree goes from general (top) to specific (bottom), and every type inherits all properties of its parents.