Angles in Polygons

Introduction

Polygons can be used to create amazing patterns. Investigating the angles inside and outside polygons can help us to understand their special qualities. In this unit we will look into the distinctive angular properties of different types of quadrilaterals and polygons.

Interior angles in quadrilaterals

If we take any four-sided

polygon

A polygon is a closed, planar figure bounded by straight line segments.polygon, we can split it into two triangles. To do this, we just join a set of opposite vertices with a

straight line

A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore one-dimensional.straight line. Fig.1 below shows how this is done in the case of a

parallelogram

A parallelogram is a quadrilateral with two sets of parallel sides.parallelogram, a

trapezium

A trapezium is a quadrilateral with one set of parallel sides and one set of non-parallel sides.trapezium, and a

rectangle

A rectangle is a quadrilateral with four interior angles of 90

.rectangle.

Figure 1.	Three quadrilaterals split into triangles.

We know that the angles in a

triangle

A triangle is a three-sided polygon.triangle add up to 180

. We can also see that, together, the angles in each pair of triangles form the angles of each

quadrilateral

A quadrilateral is a polygon with four sides.quadrilateral. So the angles in each quadrilateral must sum to 360

Figure 2.	A trapezium.

Figure 3.	A kite.

Exterior angles in quadrilaterals

Look at the quadrilateral in Fig.4 below. All of its sides have been extended and its interior (blue) and exterior (red) angles marked.

Figure 4.	The interior and exterior angles of a quadrilateral.

Notice that each exterior/interior angle pair lie on a straight line. Each pair must therefore sum to 180

.

In any quadrilateral, there will be four such pairs. So we can see that:

However, we already know that the sum of the interior angles is 360

, so:

So the exterior angles of a quadrilateral sum to 360

Figure 5.	A quadrilateral.

The parallelogram

Many engineering designs rely on the parallelogram and its properties. For instance, construction vehicles such as that below use the parallelogram shape to allow movement and stability.

Figure 6.	A parallelogram in a digger.

Using information about

parallel

Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.parallel lines,

perpendicular

Two lines or planes are perpendicular if they are at right angles to one another.perpendicular lines, and angles, it is possible to work out all the interior and exterior angles of parallelogram from just one given angle.

Figure 7.	The angles of a parallelogram.

Work out the labelled angles in Fig.7, in the order given below, using the clues. Enter each value in the table as you work it out.

Angle	Size	Clue
s		On a straight line with w.
x		Forms an alternate angle pair with s.
t		On a straight line with x.
y		Forms an alternate angle pair with t.
u		Forms a corresponding angle pair with x.
z		Forms an alternate angle pair with u.
v		Forms a corresponding angle pair with y.

What do you notice about the angles that you have entered? You should find that in the whole table, each angle has one of only two values.

If we look specifically at the interior angles s, t, u, and v, then we see that the opposite interior angles in the parallelogram are equal. Also, any two adjacent interior angles sum to 180

.

You can see these relationships between the interior angles in a parallelogram in Fig.8. Drag a side or corner to see how the angles are related.

Click on the figure below to interact with the model.

Figure 8. A parallelogram.

We can show exactly why these angle patterns exist by using the model in Fig.9 below. Click on and drag the blue angle in Fig.9. If you place the intersecting lines that form the blue angle over each corner of the parallelogram in turn you will see that each intersection is the same. We can say that the intersections are congruent.

Click on the figure below to interact with the model.

Figure 9. A parallelogram and the lines that form it.

Figure 10.	A parallelogram.

Calculate the missing angles in the parallelogram in Fig.10 above.

l	=
n	=
p	=
m	=

Interior angles in other polygons

The same methods that we use to look at quadrilaterals can be applied to other polygons, such as irregular pentagons and hexagons. In Fig.11, an irregular

octagon

An octagon is an eight-sided polygon.octagon has been divided up into triangles to help investigate its interior angles. We can divide the irregular

hexagon

A hexagon is a six-sided polygon.hexagon and the irregular

pentagon

A pentagon is a five-sided polygon.pentagon into triangles in the same way by joining vertices with lines inside the shapes only.

Click on the figure below to interact with the model.

Figure 11. Dividing polygons into triangles.

Try dragging the corners of the three shapes to see how the triangles move.

So the sum of the pentagon's interior angles is:

= 540

Likewise, the sum of the hexagon's interior angles is:

= 720

In the examples we have looked at, the number of triangles has been two fewer than the number of sides in the shape. In fact, this is the case for any polygon.

No. of sides in polygon	Number of triangles	Sum of interior angles
4	2
5	3
6	4
7	5
8	6	etc.

This expression can be rearranged as follows:

We can show this visually by dividing polygons into triangles in a different way. If we place a

point

A point has no properties except position. It is an object with zero dimensions.
Points in the x-y plane can be specified using x and y coordinates.

point anywhere inside the following hexagon, (2), we can join each

vertex

A vertex is defined as the common endpoint of two lines.

vertex to this point with a straight line (3).

Figure 12.	An irregular hexagon.

We have created 6 triangles with angles as shown (4). So the sum of the triangles' angles is

.

However, six of the triangles' angles are formed around the central point (5). These sum to 360

, but do not contribute to the interior angles of the hexagon. So here:

We can apply the same reasoning to any type of polygon. In every case, each side of the polygon forms the base of one triangle. So if n is the number of sides the polygon has, n triangles are formed.

Figure 13.	n-sided polygons divided into n triangles.

So, generally:

Exterior angles in other polygons

The exterior angles of any

convex

A polygon is convex if all of its interior angles are less than 180

.convex polygon sum to 360

. Fig.14 below illustrates one way of understanding this.

Figure 14.	Identifying the exterior angles of a polygon.

We can explain this fact mathematically, using expressions that we have studied earlier. As we saw when we looked at quadrilaterals, each of the interior and

exterior angle

When the side of a convex polygon is produced (lengthened), the exterior angle is the angle between this line and an adjacent side.exterior angle pairs adds up to 180°, as together they make up a straight line. So the total of the interior and exterior angles in a convex polygon is equal to the number of straight lines forming the polygon multiplied by 180°.

Figure 15.	The interior and exterior angles of a heptagon

So the interior and exterior angles of the above

heptagon

A heptagon is a seven-sided polygon.heptagon sum to:

So, generally, the interior and exterior angles of an n-sided polygon sum to:

Since we know a general expression for the sum of interior angles of an n-sided polygon, we can work out an expression for the exterior angle sum:

So we have found that the exterior angles of a convex polygon sum to 360

, regardless of the number of sides it has.

Summary

Splitting polygons up into triangles is a useful way of finding out about their interior angles.

The interior angles of a quadrilateral sum to 360

.

The exterior angles of convex polygon sum to 360

.

The interior angles of an n-sided polygon sum to

Exercises

Figure 16.	Angles in a hexagon.

1. For each angle labelled in Fig.16 above, state whether it is an exterior angle, an interior angle or neither, in relation to the hexagon.

a
b
c
d
e
f
g
h

Figure 17.	Calculating angles in polygons.

2. What is the missing angle in each of the shapes in Fig.17 above?

a =
b =
c =
d =

Figure 18.	Calculating angles in a parallelogram.

3. Look at the parallelogram in Fig.18 above. Match each angle to its size.

a
b
c
d

Figure 19.	A pentagon, heptagon, and octagon.

4. What is the sum of the interior angles for the three shapes shown in Fig.19 above?

Pentagon
Heptagon
Decagon

Figure 20.	A polygon.

Figure 21.	Part of a hexagon with two lines of reflection shown as dotted lines.

7. Fig.21 above represents part of a regular hexagon with two lines of

reflection

A 2-D transformation in the coordinate plane. A line of reflection must be specified and then an image is created for each point in a shape.

The line segment joining each point with its image is perpendicular to the line of reflection and bisected by it. reflection shown as dotted lines.

What is the size of angle a?
What is the size of angle b?
What is the size of angle c?

Figure 22.	Part of a regular polygon.

Figure 23.	Calculate the angle.

Figure 24.	Calculate angles.

10. The shape in Fig.24 above is an irregular hexagon with lines of

symmetry

A plane figure has symmetry if the effect of a reflection or rotation is to produce an identical-looking figure in the same position. symmetry as indicated by the dotted lines. From the information given, work out the size of angles a, b, c, and d.

a =
b =
c =
d =

Well done!

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