Polygons and PolyhedraQuadrilaterals
In the previous course we investigated many different properties of triangles. Now let’s have a look at quadrilaterals.
A regular quadrilateral is called a
A square is a quadrilateral with four equal sides and four equal angles.
For slightly “less regular” quadrilaterals, we have two options. If we just want the angles to be equal, we get a A rectangle is a quadrilateral in which all four angles are 90°. A rhombus (the plural is rhombuses or rhombi) is a quadrilateral in which all sides have the same length.
A Rectangle is a quadrilateral with four equal angles.
A Rhombus is a quadrilateral with four equal sides.
There are a few other quadrilaterals, that are even less regular but still have certain important properties:
If both pairs of opposite sides are Two or more lines are parallel if they never intersect. They have the same slope and the distance between them is always constant.
If two pairs of adjacent sides have the same length, we get a Kite.
If at least one pair of opposite sides is parallel, we get a Trapezium.
Quadrilaterals can fall into multiple of these categories. We can visualise the hierarchy of different types of quadrilaterals as a A Venn diagram visualises multiple properties or events that overlap.
For example, every rectangle is also a
To avoid any ambiguity, we usually use just the most specific type.
Now pick four points, anywhere in the grey box on the left. We can connect all of them to form a quadrilateral.
Let’s find the midpoint of each of the four sides. If we connect the midpoints, we get
Try moving the vertices of the outer quadrilateral and observe what happens to the smaller one. It looks like it is not just any quadrilateral, but always a
But why is that the case? Why should the result for any quadrilateral always end up being a parallelogram? To help us explain, we need to draw one of the A diagonal of a polygon is a line segment that connects two vertices that are not next to each other.
The diagonal splits the quadrilateral into two triangles. And now you can see that two of the sides of the inner quadrilateral are actually
In the previous course we showed that The midsegments of a triangle are the lines that connect the midpoints of different sides of the triangle.
We can do exactly the same with the second diagonal of the quadrilateral, to show that both pairs of opposite sides are parallel. And this is all we need to prove that the inner quadrilateral is a A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Parallelograms
It turns out that parallelograms have many other interesting properties, other than opposite sides being parallel. Which of the following six statements are true?
In geometry, two figures are congruent if are identical in size, shape and measure. This means we could move, flip or rotate them to exactly fit on top of each other.
An angle bisector is a line or ray that splits an angle in half, into two congruent, smaller angles.
Of course, simply “observing” these properties is not enough. To be sure that they are always true, we need to prove them:
Opposite Sides and Angles
Diagonals
Let’s try to prove that the opposite sides and angles in a parallelogram are always congruent.
Start by drawing one of the diagonals of the parallelogram.
The diagonal creates four new angles with the sides of the of the parallelogram. The two red angles and the two blue angles are Alternate angles are formed by two parallel lines which are crossed by a traversal line, and they are always congruent. In the diagram below, each of the pairs of angles labelled 1, 2, 3 and 4 are alternate. The two angles in every pair lie on a different parallel line, and opposite sides of the traversal. The angle pairs 1 and 2 are called alternate exterior angles, because they lie outside the parallel lines, and the angle pairs 3 and 4 are called alternate interior angles.
Now if we look at the two triangles created by the diagonal, we see that they have two congruent angles, and one congruent side. By the
This means that the other corresponding parts of the triangles must also be congruent: in particular, both pairs of opposite sides are congruent, and both pairs of opposite angles are congruent.
It turns out that the converse is also true: if both pairs of opposite sides (or angles) in a quadrilateral are congruent, then the quadrilateral has to be a parallelogram.
Now prove that the two diagonals in a parallelogram bisect each other.
Let’s think about the two yellow triangles generated by the diagonals:
- We have just proved that the two green sides are congruent, because they are opposite sides of a parallelogram.
- The two red angles and two blue angles are congruent, because they are
??? .
By the
Now we can use the fact the corresponding parts of congruent triangles are also congruent, to conclude that
Like before, the opposite is also true: if the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Kites
We showed above that the two pairs of
The name Kite clearly comes from its shape: it looks like the kites you can fly in the sky. However, of all the special quadrilaterals we have seen so far, the Kite is the only one that can also be A concave polygon has at least one internal angle greater than 180°. At least one of the diagonals lies outside the polygon. A common way to identify a concave polygon is to look for a “caved-in” side of the polygon. Concave is the opposite of convex polygons.

A convex kite
A concave kite that looks like an arrow
You might have noticed that all kites are If a shape has reflectional symmetry, the axis of symmetry is the line that divides it into two equal halves.
The diagonal splits the kite into two congruent triangles. We know that they are congruent from the Two triangles are congruent if their three sides have the same length. This is called the SSS congruence condition for triangles.
Using CPOCT stands for corresponding parts of congruent triangles. It means that if two triangles are congruent, then all of their corresponding components (angles, sides, midsegments, …) must also be congruent to each other.
This means, for example, that the diagonal is a
We can go even further: if we draw the other diagonal, we get two more, smaller triangles. These must also be congruent, because of the Two triangles are congruent if their three sides have the same length. This is called the SSS congruence condition for triangles.
This means that angle α must also be the same as angle β. Since they are adjacent, Two angles are supplementary if they add up to 180° (a semi-circle).
In other words, the diagonals of a kite are always
Area of Quadrilaterals
When calculating the area of triangles in the previous course, we used the trick of converting it into a
Parallelogram
Trapezium
Kite
Rhombus
On the left, try to draw a rectangle that has the same area as the parallelogram.
Can you see that the missing triangle on the left is
Area = base × height
Be careful when measuring the height of a parallelogram: it is usually not the same as one of the two sides.
Recall that trapeziums are quadrilaterals with one pair of parallel sides. These parallel sides are called the bases of the trapezium.
Like before, try to draw a rectangle that has the same area as this trapezium. Can you see how the missing and added triangles on the left and the right cancel out?
The height of this rectangle is the
The width of the rectangle is the distance between the
Like with The midsegments of a triangle are the lines that connect the midpoints of different sides of the triangle.
If we combine all of this, we get an equation for the area of a trapezium with parallel sides a and c, and height h:
In this kite, the two diagonals form the width and the height of a large rectangle that surrounds the kite.
The area of this rectangle is
This means that the area of a kite with diagonals
A A rhombus (the plural is rhombuses or rhombi) is a quadrilateral in which all sides have the same length.
This means that to find the area of a rhombus, we can use either the equation for the area of a parallelogram, or that for the area of a kite:
Area = base × height =
In different contexts, you might be given different parts of a Rhombus (sides, height, diagonals), and you should pick whichever equation is more convenient.