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9.05 Quadrilaterals

Lesson

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.

First box has 4 dots. Second box has 4 connected dots forming a quadrilateral. Third box has quadrilateral with angle arcs.

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.

This image shows the angle sum of a quadrilateral. Ask your teacher for more information.

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:

This image shows examples of convex and concave quadrilaterals. Ask your teacher for more information.

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:

Concave quadrilaterals where the internal points are highlighted and dashed lines are opposite of them forming a triangle.

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.

Concave quadrilateral with angles of 64, 23, x, and 19 degrees.
Worked Solution
Create a strategy

Add the angles and equate to 360.

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

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.

Special 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:

A trapezium with 2 parallel sides and each pair of cointerior angles between the parallel sides are supplementary.

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:

3 different parallelograms with different orientations and sizes.

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.

2 parallelograms where the first shows the opposite sides and the second shows the opposite angles have same markings.

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.

Convex and concave kites

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:

Opposite angles between the convex and concave kites have double markings.

If a parallelogram has four equal angles, they are automatically right angles. A parallelogram with four right angles is called a rectangle.

2 rectangles where the first shows 4 right angles and the second shows each pair of opposite sides have the same markings.

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:

Two rhombuses with different orientations but both show all sides of them has the same markings.

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.

Square with all of its sides with the same marking and all of its angles are right angles.

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
 A quadrilateral where two pairs of adjacent sides have the same markings.
B
 A quadrilateral where two pairs of opposite sides have the same parallel markings.
C
 A quadrilateral where two pairs of opposite sides have the same parallel markings.
D
A quadrilateral where the sides do not have any markings.
E
 A quadrilateral where two pairs of adjacent sides have the same markings.
F
A quadrilateral where the sides do not have markings.
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.

Example 3

Is the quadrilateral below a trapezium?

A quadrilateral where the angles on one side are 113 and 67 degrees.
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+113Add the angles
\displaystyle =\displaystyle 180\degreeEvaluate

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.

Idea summary

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:

This image shows the family diagram of quadrilaterals. Ask your teacher for more information.

Outcomes

VCMMG263

Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral

VCMMG262

Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral

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