Chapter 4: Quadrilaterals
Look at the room you are in. Doors are rectangles. Windows are squares or rectangles. Floor tiles are usually squares or rhombuses. The pages of this book are rectangles. Even kites you fly are special quadrilaterals. Four-sided shapes are everywhere because they are easy to make, fit together neatly, and have very useful symmetry.
A quadrilateral is any closed figure formed by four straight line segments. Within this huge family there are smaller, more famous families: parallelograms, rectangles, rhombuses, squares, trapeziums, and kites. Each member is built by adding a special property to the previous one , equal sides, parallel sides, right angles. Knowing the family relationships lets you instantly transfer a fact about parallelograms (say, "opposite angles are equal") to all its descendants.
In this chapter you will learn the angle sum property of quadrilaterals, properties of the main special quadrilaterals, and how their diagonals behave. You will also see how to compute angles using these properties without measuring, and how the same logical argument can be applied to any polygon, no matter how many sides.
By the end you will be able to look at a quadrilateral, name it correctly, and predict everything about its angles and diagonals from its name alone.
What's inside
- What is a quadrilateral? , definitions, angle sum, naming conventions.
- Special quadrilaterals: parallelograms , sides, angles, diagonals.
- Rectangle, rhombus, and square , more symmetry, more rules.
- Trapezium and kite , two more useful shapes.
- Angles in polygons , extending the angle sum to any polygon.
Key results
| Shape | Distinguishing property |
|---|---|
| Parallelogram | both pairs of opposite sides parallel |
| Rectangle | parallelogram with one right angle (hence all four) |
| Rhombus | parallelogram with all four sides equal |
| Square | rectangle AND rhombus |
| Trapezium | exactly one pair of opposite sides parallel |
| Kite | two pairs of adjacent sides equal |
Angle sums
| Property | Formula |
|---|---|
| Sum of interior angles of a quadrilateral | |
| Sum of interior angles of an -sided polygon | |
| Sum of exterior angles of any polygon |
Parallelogram quick facts
- Opposite sides equal.
- Opposite angles equal.
- Adjacent angles are supplementary ( each pair).
- Diagonals bisect each other.
Rectangle: parallelogram + diagonals are equal. Rhombus: parallelogram + diagonals are perpendicular. Square: rectangle + rhombus. Kite: one diagonal bisects the other at right angles.
How to read this chapter
Use a notebook, a ruler, and a protractor. Most properties become "obvious" once you actually draw the shape carefully and measure. Try to prove each property from a previous one rather than just memorising , that is how mathematicians think.