7. Geometry

Lesson

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^\circ$360°.

Did you know?

Mathematical language can be confusing sometimes. A shape with three sides is called a **triangle**, because it has three ("tri") angles ("angle"). A shape with four sides is called a **quadrilateral**, because it has four ("quad") sides ("lateral"). A shape with five sides is called a **pentagon**, because it has five ("penta") sides ("gon"). Some of these words come from Latin, and others from Ancient Greek. It would be much easier if we followed the same pattern!

- Following the Greek style, we would have a
**trigon**, a**tetragon**, and a**pentagon** - Following the Latin style, we would have a
**triangle**, a**quadrangle**, and a**quintangle** - ... or a
**trilateral**,**quadrilateral**, and**quintilateral**

But mathematical language is just like any other language - full of inconsistencies and historical relics, developed over time by different cultures and peoples for different reasons. But what matters most of all are the geometrical concepts that these words describe, transcending both culture and language alike.

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.

If two sides of a quadrilateral are parallel, we call it a trapezoid. 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 trapezoid 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 trapezoid 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. Here are two more shapes that are also parallelograms:

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 trapezoid 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, and has these properties:

- All angles are right angles (by definition)
- Opposite sides are parallel and equal in length (because it is a parallelogram)

Here are two rectangles:

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 trapezoid)
- 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 trapezoid! These are the properties that all squares have:

- All sides are equal in length (because it is a rhombus)
- All angles are right angles (because it is a rectangle)

Here is a square:

Summary

The properties of a quadrilateral are inherited from one another. These properties are summarised in this diagram:

Select the two parallelograms:

- ABCDEF

Is the quadrilateral below a trapezoid?

Yes

ANo

B

Solve for the value of $x$`x` in the diagram below.

Enter each line of working as an equation.

Solve problems involving angle properties, including the properties of intersecting and parallel lines and of polygons.