Just like we did with triangles , we can take any four points that do not lie on a line and join them with four segments to form a shape. We call this shape a quadrilateral.
There is an enormous variety of four-sided shapes. The one thing they all have in common is that they can always be split down the middle to make two triangles. This means that: the angle sum of a quadrilateral is 360\degree.
The two triangles each form a straight angle, and together they form a full revolution.
We will explore the different kinds of quadrilaterals and the families they are organised into.
The first division lies between the convex quadrilaterals and the concave quadrilaterals:
Notice that the concave quadrilaterals stick into themselves - we can form a triangle with three of their points, with the fourth one lying inside it:
Most of the time we will be looking at convex quadrilaterals. With one exception, all the special quadrilaterals we mention in this lesson are convex.
Solve for the value of x in the diagram below.
A quadrilateral is any shape that has four sides. The sum of its angles is 360\degree.
Concave quadrilaterals can form a triangle with its three points which makes it different to convex quadrilaterals.
If two sides of a quadrilateral are parallel, we call it a trapezium. The angles on the sides connecting the parallel lines are always supplementary, because they form a cointerior angle pair:
If the four sides of a quadrilateral form two pairs of parallel sides, we have a special kind of trapezium called a parallelogram. Here are some parallelograms:
These shapes always have all of these properties:
Opposite sides are parallel (by definition)
Consecutive angles are supplementary (because it is a trapezium in two ways)
Opposite angles are equal
Opposite sides are equal in length
If a quadrilateral has any one of these properties, it will be a parallelogram with all the other properties as well.
A different kind of quadrilateral is called a kite, where the shape has two pairs of adjacent sides that are equal in length. Unlike the trapezium family, kites can be both concave and convex. Sometimes you might hear a concave kite referred to as a dart.
If a shape is a kite, it has an additional property. The angles between each pair of equal sides may be different, but the other two angles are always the same:
If a parallelogram has four equal angles, they are automatically right angles. A parallelogram with four right angles is called a rectangle.
A different kind of shape is both a parallelogram and a kite at the same time, called a rhombus, which has these properties:
Opposite sides are parallel (because it is a parallelogram)
Consecutive angles are supplementary (because it is a trapezium)
Opposite angles are equal (because it is a parallelogram)
All sides are equal in length, because:
Opposite sides are equal in length (parallelogram)
Two pairs of adjacent sides that are equal in length (kite)
Here are two rhombuses:
The most special kind of quadrilateral is the square. It is a combination of a rectangle and a rhombus, so it is also a parallelogram, a kite, and a trapezium.
Select all the parallelograms:
Is the quadrilateral below a trapezium?
Trapeziums have one pair of parallel sides and the angles on the sides connecting the parallel sides are supplementary.
Parallelograms have two pairs of parallel sides and their opposite angles are equal.
Kites have two pairs of equal adjacent sides.
One pair of opposite angles in a kite are equal.
The properties of a quadrilateral are inherited from one another. These properties are summarised in this diagram: