12.06 Quadrilaterals

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:

The internal point has been highlighted for each concave quadrilateral.

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.

Examples

Example 1

Solve for the value of x in the diagram below.

Worked Solution

Create a strategy

Add the angles and equate to 360.

Apply the idea

\displaystyle x+64+23+19	\displaystyle =	\displaystyle 360	Add the angles and equate to 360
\displaystyle x+106	\displaystyle =	\displaystyle 360	Evaluate the addition
\displaystyle x+106-106	\displaystyle =	\displaystyle 360-106	Subtract 106 from both sides
\displaystyle x	\displaystyle =	\displaystyle 254	Evaluate

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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.

Here are two more shapes that are also parallelograms.

Because these shapes have one of the above properties, they have all of them - including the defining one, that opposite sides are parallel.

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.

Rectangles have these properties:

• All angles are right angles (by definition)

• Opposite sides are parallel and equal in length (because it is a parallelogram)

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.

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)

Examples

Example 2

Select all the parallelograms:

A

B

C

D

E

F

Worked Solution

Create a strategy

Recall that any shape that has two pairs of parallel sides is a parallelogram.

Apply the idea

The answers are options B and C.

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Example 3

Is the quadrilateral below a trapezium?

A

Yes

B

No

Worked Solution

Create a strategy

Add the angles to check if its sum is 180\degree to tell that they are supplementary.

Apply the idea

\displaystyle \text{Sum of angles}	\displaystyle =	\displaystyle 67+113	Add the angles
	\displaystyle =	\displaystyle 180\degree	Evaluate

Since the sum of the angles that are on the same side is 180\degree, they are supplementary, which makes the opposite sides parallel.

So, the quadrilateral is a trapezium. The answer is option A.

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