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.
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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.
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In Fig.2 above, what type of triangle is triangle ABD?
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Complete the sentences below.

  • Line AD is parallel to line . Therefore, angle is the same size as angle DBC since they are angles.
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What is the size of
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
ABD?
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So what is the size of angle ABC?
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Figure 3.   A kite.
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In Fig.3 above, what type of triangle are both ABD and BCD?
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Use your answer to the question above to complete the following sentence:

  • Angle , and angle .
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What is the size of ?
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What is the size of ?
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Use your last two answers to find the missing value.
  • .
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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.
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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.
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What is the value of x in Fig.5 above?
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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.
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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.
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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.

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

  • l =
    n =
    p =
    m =

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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.

How many triangles make up the pentagon in Fig.11?
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How many triangles make up the hexagon?
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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.

If n is the number of sides a polygon has, complete this
expression
In mathematics, an expression is combination of known symbols including variables, constants and/or numbers. Algebraic equations are made up of expressions. Examples of expressions include : , , and .
expression
for the sum of its interior angles.
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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.
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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.
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So, generally:



What is the sum of the interior angles in an octagon?
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What is the sum of the interior angles in a 22-sided polygon?
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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.
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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
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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.

If all the exterior angles in a
dodecagon
A dodecagon is a 12-sided polygon.
dodecagon
are equal, how many degrees does each exterior angle measure?
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So what is the value of each
interior angle
An interior angle is the angle between adjacent sides at a vertex of a polygon.
interior angle
in the dodecagon?
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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.
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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
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Figure 17.   Calculating angles in polygons.
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2. What is the missing angle in each of the shapes in Fig.17 above?

  • a =
    b =
    c =
    d =

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Figure 18.   Calculating angles in a parallelogram.
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3. Look at the parallelogram in Fig.18 above. Match each angle to its size.
  • a
    b
    c
    d
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Figure 19.   A pentagon, heptagon, and octagon.
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4. What is the sum of the interior angles for the three shapes shown in Fig.19 above?

  • Pentagon
    Heptagon
    Decagon

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Figure 20.   A polygon.
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5. What is the shape in Fig.20 called?
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6. The angles p and q in Fig.20 above are equal in size because...
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Figure 21.   Part of a hexagon with two lines of reflection shown as dotted lines.
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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?

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Figure 22.   Part of a regular polygon.
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8. In Fig.22 above you can see the edge of a
regular polygon
A regular polygon has sides of equal length and interior angles of the same size.
regular polygon
. How many sides does it have?
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Figure 23.   Calculate the angle.
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9. What is the size of the internal angle s in the rhombus in Fig.23 above?
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Figure 24.   Calculate angles.
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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 =

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Well done!
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