Square, rectangle, rhombus, parallelogram, trapezium, kite. Six names, and most students learn them as six separate definitions to memorise. They are not separate. They are one family tree, and each name is the one above it with one extra condition added. Learn the tree and the definitions come for free.
The tree
| Shape | What makes it that shape | Built from |
|---|---|---|
| Quadrilateral | Any closed shape with four straight sides | The starting point |
| Trapezium | One pair of opposite sides parallel | Quadrilateral + one parallel pair |
| Parallelogram | Both pairs of opposite sides parallel | Trapezium + a second parallel pair |
| Rhombus | A parallelogram with all four sides equal | Parallelogram + equal sides |
| Rectangle | A parallelogram with all four angles 90° | Parallelogram + right angles |
| Square | All four sides equal AND all four angles 90° | Rhombus + right angles, or rectangle + equal sides |
| Kite | Two pairs of equal sides that are next to each other, not opposite | Off to one side of the tree |
A square is a rectangle. It is also a rhombus, and a parallelogram, and a trapezium, and a quadrilateral. All of those at once. Being a square does not stop it being the others, in the same way that being a sparrow does not stop something being a bird. Students resist this because in ordinary speech we say a shape is either a square or a rectangle. In mathematics, going down the tree adds conditions and never removes any.
What the parallel sides force to happen
Every property of a parallelogram comes out of the last topic in Unit 1, where you learned that parallel lines slide an angle along unchanged. Draw a parallelogram and treat one side as a transversal cutting the two parallel sides. Alternate angles are equal, so the two opposite angles must be equal. Co-interior angles add to 180, so any two angles next to each other add to 180. Nothing here has to be memorised separately.
| Property of a parallelogram | Where it comes from |
|---|---|
| Opposite angles are equal | Alternate angles on parallel lines |
| Neighbouring angles add to 180° | Co-interior angles on parallel lines |
| Opposite sides are equal | A diagonal splits it into two identical triangles |
| The diagonals cut each other in half | Also from those identical triangles |
| All four angles add to 360° | True for every quadrilateral, not just this one |
Draw one diagonal across any four sided shape. It splits into two triangles, and nothing has been added or lost. Each triangle contributes 180, so the four corners of the quadrilateral together must be 2 times 180, which is 360. This works for a square, for a kite, for a squashed irregular shape, for anything with four straight sides.
In a parallelogram the diagonals cut each other in half but are not equal and do not meet at right angles. In a rectangle they are also equal. In a rhombus they also cross at right angles. In a square all three things are true at once, because a square is both a rectangle and a rhombus.
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Three angles of a quadrilateral are 85°, 100° and 65°. Find the fourth.
One angle of a parallelogram is 70°. Find the other three.
| Angle | Reason | Value |
|---|---|---|
| A | Given | 70° |
| B | Neighbouring angles are co-interior on parallel sides, so they add to 180 | 110° |
| C | Opposite to A, and opposite angles of a parallelogram are equal | 70° |
| D | Opposite to B | 110° |
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The quadrilaterals form a family tree. Each name is the one above it with one extra condition.
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Question 1 of 10
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