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 . All of its sides have the same length, and all of its angles are equal.

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 rectangle. If we just want the sides to be equal, we get a rhombus.

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 parallel, we get a Parallelogram.

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 Venn diagram:

For example, every rectangle is also a , and every is also a kite. A rhombus is a square and a rectangle is a trapezium.

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 diagonals of the original quadrilateral.

The diagonal splits the quadrilateral into two triangles. And now you can see that two of the sides of the inner quadrilateral are actually of these triangles.

In the previous course we showed that midsegments of a triangle are always parallel to its base. In this case, it means that both these sides are parallel to the diagonal – therefore they must also be .

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 parallelogram.

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?

The opposite sides are congruent.

The internal angles are always less than 90°.

The diagonals bisect the internal angles.

The opposite angles are congruent.

Both diagonals are congruent.

Adjacent sides have the same length

The two diagonals bisect each other in the middle.

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

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, so they must each be .

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 congruence condition, both triangles must be congruent.

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.

Diagonals

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 condition, both of the yellow triangles must therefore also be congruent.

Now we can use the fact the corresponding parts of congruent triangles are also congruent, to conclude that AM‾ = CM‾ and BM‾ = DM‾. In other words, the two diagonals intersect at their midpoints.

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 sides of a parallelogram are congruent. In a kite, two pairs of adjacent sides are congruent.

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 concave: if it is shaped like a dart or arrow:

A convex kite

A concave kite that looks like an arrow

You might have noticed that all kites are . The axis of symmetry is .

The diagonal splits the kite into two congruent triangles. We know that they are congruent from the SSS condition: both triangles have three congruent sides (red, green and blue).

Using CPOCT, we therefore know that the corresponding angles must also be congruent.

This means, for example, that the diagonal is a of the two angles at its ends.

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 SAS condition: they have the same two sides and included angle.

This means that angle α must also be the same as angle β. Since they are adjacent, supplementary angles both α and β must be °.

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 . It turns out that we can also do that for some quadrilaterals:

Parallelogram

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 the overlapping triangle on the right? Therefore the area of a parallelogram 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.

Trapezium

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 parallel sides of the trapezium.

The width of the rectangle is the distance between the of the two non-parallel sides of the trapezium. This is called the midsegment of the trapezium.

Like with triangles, the midsegment of a trapezium is its two bases. The length of the midsegment is the average of the lengths of the bases: a+c2.

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:

A=h×a+c2

Kite

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 the area of the kite. Can you see how each of the four triangles that make up the kite are the same as the four gaps outside it?

This means that the area of a kite with diagonals d1 and d2 is

Area=12d1×d2.

Rhombus

A Rhombus is a quadrilateral that has four congruent sides. You might remember that every rhombus is a – and also a .

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 = 12 d1 × d2.

In different contexts, you might be given different parts of a Rhombus (sides, height, diagonals), and you should pick whichever equation is more convenient.

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