Download Part 1 Interior and exterior angles

Mathematics Stage 5
SGS5.2.1 Properties of geometrical figures
Part 1
Interior and exterior angles
Number: 43688
Title: SGS5.2.1 Properties of Geometrical Figures
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Extracts from Mathematics Syllabus Years 7-10 © Board of Studies, NSW 2002
Unit Overview p iiiiv ; Part 1 p 3
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Contents – Part 1
Introduction – Part 1 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz.................................................................5
Interior angles in polygons ................................................9
Regular polygons ............................................................17
Exterior angles of a polygon............................................23
Walking a fine line ...........................................................31
Suggested answers – Part 1 ...........................................37
Exercises – Part 1 ...........................................................49
Part 1
Interior and exterior angles
1
2
SGS5.2.1 Properties of geometrical angles
Introduction – Part 1
In this part you will develop rules for finding the sum of angles both
inside and outside different shapes called polygons. You will also learn,
through practical tasks, why the sum of the exterior (outside) angles of
any polygon is always the same total. This part also enables you to
practise solving problems using these rules.
Indicators
By the end of Part 1, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
calculating the interior angle sum of polygons by dissecting them
into triangles
•
applying the result for the interior angle sum of a triangle in other
shapes
•
defining the exterior angle of a polygon
•
establishing that the exterior angle sum of all convex polygons is
360 0
•
applying rules to find unknown angles in polygons.
By the end of Part 1, you will have been given the opportunity to work
mathematically by:
•
expressing the interior angle sum of a polygon with n sides as
(n − 2) ×180 0
•
calculating the size of interior and exterior angles in any regular
polygon
•
solving problems using angle sum of polygon results.
Source:
Part 1
Adapted from outcomes of the Mathematics Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_
710_syllabus.pdf > (accessed 04 November 2003).
© Board of Studies NSW, 2002.
Interior and exterior angles
3
4
SGS5.2.1 Properties of geometrical angles
Preliminary quiz
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1
a
What is the name of the angle below? ____________________
2
Answer questions about the following diagram.
42° x°
a
Explain why x o + 42 0 = 180 0 ? ___________________________
___________________________________________________
b
Calculate the value of x o . ______________________________
___________________________________________________
c
Choose the correct word from the list, and write it in the
sentence below.
Word list: complement, complementary, supplement,
supplementary
20 0 is the ___________________________________ of 160 0 .
Part 1
Interior and exterior angles
5
3
Calculate the size of ∠ABC in the triangle below.
A
69°
B
56°
C
_______________________________________________________
_______________________________________________________
4
A square is a regular shape. However, a rhombus and a rectangle
are not.
square
a
rhombus
rectangle
What is it that makes a shape regular? ____________________
___________________________________________________
6
SGS5.2.1 Properties of geometrical angles
5
Name these shapes, by using the word ‘regular’ or ‘irregular’,
together with the word that describes their number of sides.
a
___________________________________________________
b
___________________________________________________
6
How many sides does a hexagon have? _______________________
7
Use the tessellated squares below to answer the following questions.
2 1
3 4
a
Part 1
Circle the correct answers.
i
Angles 1, 2, 3 and 4 combine together to form a: right
angle, revolution, straight angle.
ii
The four angles above have a sum of: 90 0 , 180 0 , 360 0 .
Interior and exterior angles
7
b
Explain why all the angles 1, 2, 3 and 4 are found by doing the
following calculation.
360 0 ÷ 4 = 90 0
___________________________________________________
c
Complete the sentence below:
Each angle of a square is equal to __________. A square
tessellates because a __________________ ( 360 0 ) is divisible
by 90 0 .
Check your response by going to the suggested answers section.
8
SGS5.2.1 Properties of geometrical angles
Interior angles in polygons
Polygons are shapes with three or more angles. The word ‘polygon’
means ‘many angles’ from the Greek ‘poly’ meaning ‘many’ and ‘gonon’
or ‘gonea’ meaning angle.
The word ‘polygon’ will refer to convex polygons unless otherwise
stated for the rest of this unit.
convex
polygon
non-convex
polygon
Polygons that are not convex are called non-convex polygons.
The table below lists the polygons that are used most frequently.
Number of angles
polygon
3
triangle
4
quadrilateral
5
pentagon
6
hexagon
8
octagon
9
nonagon
10
decagon
12
dodecagon
If a polygon has three angles, (and thus three sides), then it is called a
triangle rather than a polygon.
The names of polygons are linked to Greek and Latin words.
For example, a decagon has ten angles inside it just like a decade has ten
Part 1
Interior and exterior angles
9
years inside it. A dozen eggs means twelve eggs. Similarly a dodecagon
has twelve sides. (This word comes from the Greek word ‘dodeka’
which actually means ‘two, ten’) Our 50 cent coin is a dodecagon.
Other words include ‘bi’ which means ‘two’ so bicycles have two wheels
and ‘tria’ which means ‘three’ so tricycles have three wheels.
Interior angles are angles inside the corners of a shape.
For example, the sum of the interior angles of a triangle is 180 0 . So, in
the figure below, ∠A + ∠B + ∠C = 180 0 .
62°
68°
50°
If in doubt, check that the angle sum of this triangle is 180 0 by using a
protractor or by adding the size of the three angles together.
Alternatively, you could draw any triangle on a piece of paper and rip the
three angles out of the triangle. If you place these three angles next to
each other you always get a straight angle of 180 0 like this.
62°
68°
50°
Note: the word ‘angle’ in ‘angle sum of a triangle is 180 0 ’ refers to the
inside (interior) angles.
Remember that the angle sum of a quadrilateral is 360 0 .
10
SGS5.2.1 Properties of geometrical angles
This can easily be shown to be true by drawing a diagonal through a
quadrilateral.
180°
180°
Quadrilaterals have two triangles inside them, each with an angle sum of
180 0 . So, the total of all the angles inside the vertices (corners) of the
quadrilateral is 360 0 because 180 0 +180 0 = 360 0 .
This process of dividing a shape into triangles can be extended further to
polygons of any size. So if you add another triangle to the quadrilateral
above, you will get this picture below.
180°
180°
How many sides does this polygon have? What is its name?
It has five sides, so it must be a pentagon.
There are three triangles inside this pentagon.
The extra triangle has an angle sum of 180 0 , so the angle sum of a
pentagon must be:
Part 1
Interior and exterior angles
11
180 0 +180 0 +180 0 = 360 0 +180 0
= 540 0.
The facts discussed above are summarised in the first three rows of the
table below.
Name of
polygon
Number of
sides
Number of
triangles
inside
polygon
Calculation for
sum of interior
angles.
Angle sum of
polygon
triangle
3
1
1×180 0
180 0
quadrilateral
4
2
2 ×180 0
360 0
pentagon
5
3
3 ×180 0
540 0
heptagon
7
decagon
99-agon
n-agon
Use the activity below to help you complete the rest of this table.
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SGS5.2.1 Properties of geometrical angles
Activity – Interior angles in a polygon
Try these.
1
Draw diagonals from vertex A in the hexagon below.
A
a
How many triangles are there inside this hexagon? __________
b
0
Complete: angle sum of a hexagon = __×180
= 720 0
c
In the table above, complete the row below ‘pentagon’, for this
hexagon.
2
A heptagon has seven angles. Its interior angle sum is 900 0 .
One number in the calculation below is incorrect. Cross it out and
replace it with the correct number.
Angle sum of a heptagon = 7 ×180 0
= 900 0
Complete the row for heptagon, by writing in the correct numbers.
Part 1
Interior and exterior angles
13
3
Draw an octagon below. (Make sure it is a convex polygon.)
Choose one vertex of your octagon and label it A. Follow the same
procedure as above by drawing diagonals from point A.
a
How many triangles inside this octagon? __________________
b
i
In the table above, write in the row below ‘heptagon’.
Write all the information you have about this octagon,
except for the last entry in the last column.
ii
Use the space below to calculate the sum of the interior
angles of this octagon.
________________________________________________
________________________________________________
iii Write this sum in the last column for the octagon.
4
Analyse the numbers in the table that you have completed so far.
What pattern can you see when you compare the number of sides of
a polygon with the number of triangles inside it?
_______________________________________________________
_______________________________________________________
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SGS5.2.1 Properties of geometrical angles
5
Show, by drawing a convex nonagon with triangles inside it, and by
written explanation, that the angle sum of a nonagon is:
angle sum of a nonagon = (9 − 2) ×180 0
= 7 ×180 0
= 1260 0
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Write in the table the information you have above for a nonagon.
Use the row above ‘decagon’ to do this.
6
Without drawing a dodecagon, complete the following for this
twelve-sided figure:
angle sum of a dodecagon = (__− 2) ×180 0
= _______________
= _______________
Write in the table the information you have for a dodecagon.
Use the row above ‘99-agon’ to do this.
7
Complete the first and the last line of this calculation for the angle
sum of a 99-sided polygon.
__________________________ = _________________
= 97 ×180 0
= ____________
Now complete the second last row of the table.
Part 1
Interior and exterior angles
15
8
Look at the last row of the table. This polygon has n sides, so we
call it an n-agon. Write an algebraic expression, using the letter n
to show how you can find the interior angle sum of a polygon with n
sides.
_______________________________________________________
Complete the last row of the table.
9
Complete the row for a decagon.
Check your response by going to the suggested answers section.
You do not need to memorise the angle sums of these polygons because
all you have to do is subtract two from the number of sides of the
polygon and multiply this result by 180.
There are three websites below. One of the sites reviews what you have
already learnt and gives two further examples. Another site lets you
explore the interior angle sum of a quadrilateral and a pentagon when
you drag points to create a new shape. The third site has multiple choice
questions that you can answer.
Access these sites about the sum of interior angles and exploring interior
angles by visiting the CLI webpage <http://www.cli.nsw.edu.au/Kto12>.
Select Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5 Space and geometry then select SGS5.2.1 Properties of
geometrical figures, Part 1.
You have developed a rule for calculating the interior angle sum of any
polygon. Now check that you can use this rule to solve these kinds of
problems by yourself.
Go to the exercises section and complete Exercise 1.1 – Interior angles in a
polygon.
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SGS5.2.1 Properties of geometrical angles
Regular polygons
Regular polygons have equal sides and equal angles. This means that if
you know the interior angle sum of a regular polygon, you can find the
size of every angle. In the simplest regular polygon, the equilateral
triangle, all the angles are the same size, that is, 60 0 . Why?
The angle sum of this triangle is 180 0 . It is regular, so the three angles
are equal.
So 180 0 ÷ 3 = 60 0
60°
60°
60°
In the activity below you will calculate the size of every angle inside a
regular polygon.
Part 1
Interior and exterior angles
17
Activity – Regular polygons
Try these.
1
Answer the following questions about the polygon below.
x°
C
a
This shape has a name that describes it exactly. What is it?
___________________________________________________
b
How many angles in this shape are the same size? ___________
___________________________________________________
c
Draw diagonals in the shape above, from vertex C.
How many triangles are there inside the shape? _____________
d
i
One triangle has an angle sum of 180 0 .
Use this fact to calculate the angle sum of the shape above.
Show your working below.
________________________________________________
________________________________________________
ii
To find the size of every angle above, you must divide this
sum by a certain number. What is this number?
________________________________________________
iii Calculate x o .
________________________________________________
________________________________________________
18
SGS5.2.1 Properties of geometrical angles
Check your response by going to the suggested answers section.
If you know the name of a regular polygon, then you also know the size
of every angle inside this polygon.
The following example helps you to understand how to write reasons to
explain your calculations when finding angles in polygons.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Calculate the value of x o in the following diagrams, giving
reasons.
a
x°
b
x°
Part 1
Interior and exterior angles
19
Solution
a
This is a pentagon so n = 5.
Angle sum of a pentagon = (n − 2) ×180 0
= (5 − 2) ×180 0
= 3 ×180 0
= 540 0.
It is regular, so all the interior angles are the same.
You can write your calculation and reason like this:
x o = 540 ÷ 5 (angle sum of a regular pentagon)
= 108 0
b
This polygon has 10 sides. It is a regular decagon.
Angle sum of a decagon = (n − 2) ×180 0
= (10 − 2) ×180
= 8 ×180
= 1440 0.
The angle next to x o in the decagon needs to be labelled.
You could label it y o or ∠ABC like this:
A
y°
B
x°
C
x°
Using the second diagram, you can now write your
solution like this:
∠ABC = 1440 ÷10 (angle sum of a regular decagon)
=144 o
x o = 180 −144 (a straight angle)
= 36 o
It is important not to get confused between the number of triangles inside
a polygon and its number of sides.
20
SGS5.2.1 Properties of geometrical angles
There are two websites below. One of the sites reviews what you have
learnt so far about regular polygons and gives two more examples one of
which extends your understanding. The other site has multiple choice
questions that you can answer.
Access related sites called ‘Each exterior angle’ by visiting the CLI
webpage <http://www.cli.nsw.edu.au/Kto12>.
Select Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5 Space and geometry and then SGS5.2.1 Properties of
geometrical figures, Part 1.
The exercise below will enable you to practise finding the size of every
angle inside regular polygons.
Go to the exercises section and complete Exercise 1.2 – Regular polygons
Part 1
Interior and exterior angles
21
22
SGS5.2.1 Properties of geometrical angles
Exterior angles of a polygon
Exterior angles are on the outside of a polygon. They are between the
side of a polygon and the extension of the other side close to it.
They look like this:
exterior angle
exterior angle
The exterior angles that you will be investigating in the next activity are
always in the same position when the polygon is rotated. For example, in
a triangle, the three exterior angles that you will be using for this activity
are like this:
not like this:
Part 1
Interior and exterior angles
23
Activity – Exterior angles of a polygon
Try these.
1
Answer the following questions about the diagram below.
50°
30° x°
100°
a
Why does 30 o + x o = 180 o ?______________________________
b
Calculate x o and write your answer in the correct place on the
triangle above. ______________________________________
c
Calculate the size of the other two exterior angles using this
same rule.
___________________________________________________
___________________________________________________
Write your answers in the correct place on the triangle above.
d
i
Add up the three exterior angles.
________________________________________________
________________________________________________
________________________________________________
ii
Complete: the sum of the exterior angles of a triangle is
________________________________________________
24
SGS5.2.1 Properties of geometrical angles
2
a
Finish drawing the other three exterior angles of the irregular
quadrilateral below. (Remember that these exterior angles
must be in the same position when you rotate the quadrilateral.)
120°
35°
60°
95°
110°
Calculate the size of the three exterior angles you have drawn.
___________________________________________________
___________________________________________________
___________________________________________________
b
i
Add up the four exterior angles.
________________________________________________
________________________________________________
________________________________________________
ii
Complete: the sum of the exterior angles of a quadrilateral
is ______________________________________________
iii Compare this answer with your previous answer for the
triangle above. ___________________________________
________________________________________________
Part 1
Interior and exterior angles
25
3
Answer questions about the irregular pentagon below.
120°
150°
60°
110°
a
100°
Check that the angle sum of the interior angles is 540 0 .
___________________________________________________
b
Draw all the exterior angles on this irregular pentagon above.
(Remember that once you draw the first exterior angle, the other
exterior angles, when you rotate the pentagon, must face the
same direction as the first.)
Write the size of each exterior angle on the diagram above.
Use the space below to do your working, or use a calculator.
c
i
Use a calculator to find the sum of the exterior angles of
this irregular pentagon. Write your answer here._________
ii
Complete: the _________ of the exterior angles of a ______
is __________.
iii Compare this answer with your previous answers for the
triangle and quadrilateral above. _____________________
d
26
The result above is true for all convex polygons. Complete: the
sum of the exterior angles of any convex polygon is always
_________. (Strange, but true!)
SGS5.2.1 Properties of geometrical angles
4
Answer the following questions to show this rule is true for the
regular octagon below.
a
Use the formula: angle sum of a polygon = (n − 2) ×180 0 to
calculate the interior angle sum of this octagon.
___________________________________________________
___________________________________________________
b
Calculate the size of each of the interior angles of this octagon.
(Remember that they are all the same.)
___________________________________________________
c
Calculate the size of each of the exterior angles of this octagon.
(Remember that they are all the same.)
___________________________________________________
d
Is the sum of these exterior angles also equal to 360 0 ? _______
Check your response by going to the suggested answers section.
The sum of the exterior angles of a polygon is 360 0 . It is a very
easy number to remember because it is the same number for all
convex polygons.
Part 1
Interior and exterior angles
27
The following example shows you how you can solve problems using
this rule.
Follow through the steps in this example. Do your own working in the
margin if you wish.
a
Calculate the value of x o .
42°
36°
61°
x°
b
84°
Calculate the size of one of the exterior angles of a regular
20-sided convex polygon.
Solution
a
This polygon has six exterior angles, so always check that
you have not left out any numbers when adding them
together.
The sum of the exterior angles of a polygon = 360 0 .
So, starting from x o and moving around the shape in a
clockwise direction, you will get:
x o + 61o + 90 o + 42 o + 36 o + 84 o = 360 o
x o + 313o = 360 o
x o = 360 o − 313o
= 47 o
If you don’t want to write an equation, then simply add the
five numbers in the diagram (including the right angle) and
then subtract this answer from 360 0 .
28
SGS5.2.1 Properties of geometrical angles
b
A 20-sided regular polygon has 20 equal exterior angles.
The total of all these angles is 360 0 , so to find one of these
equal angles you must divide 360 by 20.
Exterior angle = Exterior angle sum ÷ number of angles
= 360 ÷ 20
= 18 o
No matter which convex polygon, regular or irregular, the exterior angle
sum is always 360 0 . If it is regular you can calculate the size of each
exterior angle by simply dividing 360 0 by the number of angles.
There are two websites below. One website reviews concepts about the
exterior angle sum of polygons and the other gives you multiple choice
questions you can answer.
Access these sites about exterior angles by visiting the CLI webpage
<http://www.cli.nsw.edu.au/Kto12>.
Select Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5 Space and geometry then select SGS5.2.1 Properties of
geometrical figures, Part 1.
Use the rule that the exterior angle sum of a polygon is 360 0 in the
following exercise.
Go to the exercises section and complete Exercise 1.3 – Exterior angles of
a polygon.
Part 1
Interior and exterior angles
29
30
SGS5.2.1 Properties of geometrical angles
Walking a fine line
What is the angle sum of the exterior
angles of a polygon?
How many degrees in a revolution?
The answer to both questions is the same number, 360 0 . They have the
same answer because the two ideas are closely linked. To understand
this link you will need to stand up and walk around using the following
two practical tasks. The first task reviews a revolution. This is a simple
task but it will help you perform the second task.
Activity – Walking a fine line
Choose a flat open space to walk around. We don’t want you crashing
into or tripping over anything while you are walking.
To review a revolution, stand up and face an object, such as a tree or a
door or a wall. Put your arm straight out in front of you, keeping it
parallel to the ground. (You may point at the object if it helps.)
Turn your body (and arm) a quarter turn. As you move, your arm sweeps
through 90 0 .
Turn another quarter turn. Your arm sweeps through a further 90 0 .
1
What is the total number of degrees that your arm has swept
through? _______________________________________________
Turn another quarter turn, with your arm outstretched.
2
Calculate the total number of degrees your arm has swept through.
_______________________________________________________
Part 1
Interior and exterior angles
31
3
a
Where are you facing when you turn another quarter turn?
___________________________________________________
b
Through how many degrees does your arm sweep when you
start and finish in the same position, looking in the same
direction? ___________________________________________
Check your response by going to the suggested answers section.
Practise the task above, until you understand the position of the right
angles that you have swept through.
In the second task you are going to walk along an imaginary rectangle.
If you have difficulty with this, imagine that you are walking around the
outside of a rectangular building. Alternatively, if you are able to, you
can draw the rectangle on the ground.
The rectangle looks like this:
object that locates your
starting direction
3 steps
3 steps
2 steps
2 steps
start here
This following practical task is a little like orienteering. Orienteering is a
sport where you must follow a map and a set of instructions to move
around a course. In orienteering, both the distance and angle that you
turn, is important. If you do not follow the instructions correctly in
orienteering you will get lost or not get to the correct checkpoints or
control markers.
However, in this task the number of steps is not important because you
are going to be concentrating on the angles that your arm sweeps
32
SGS5.2.1 Properties of geometrical angles
through. The steps are there only to give you a guide. You can walk
kilometres if you wish!
Activity – Walking a fine line
Standing with your arm outstretched in front of you and facing the same
object, take three steps, finishing by putting both feet together.
4
Has your arm swept through any angle yet? ____________________
Now turn, with your arm outstretched so that you are ready to walk along
the next side of the rectangle. (Do not walk, just turn.)
5
Choose the angle that your arm has swept through.
a
b
d
e
c
_______________________________________________________
6
Draw the angle that your arm has swept through on the
rectangle above.
Walk two steps forward, finishing by putting both feet together.
With your arm outstretched again, turn so that you are ready to walk
along the next side of the rectangle.
Part 1
Interior and exterior angles
33
7
Draw the angle that your arm has swept through on the corner below.
8
Draw this angle on the rectangle above.
Walk three steps and turn.
9
Draw the angle that your arm sweeps through on the rectangle
above.
Walk two steps.
10 You have finished walking along the rectangle, but are you facing
the same object? ________________________________________
Finish by facing in the same direction that you started.
11 Draw the final angle that your arm sweeps through on the rectangle
above.
12 Compare all four angles that you have drawn on your rectangle by
answering the following questions.
a
The size of all four angles are the same.
i
How many degrees are they each? ___________________
ii
What is their total? _______________________________
b
Are all these angles outside the rectangle? _________________
c
Rotate the rectangle and look at each angle in the same position.
Are all the angles facing the same direction? _______________
d
34
Complete: all four angles are the _________________ angles of
the rectangle.
SGS5.2.1 Properties of geometrical angles
e
Complete: the total amount of turning while walking around
this rectangle is ____ 0 . This is the same as the ___________ of
the exterior angles of a convex _________________ and the
same as a full rotation which is called a _________________.
Check your response by going to the suggested answers section.
If you start and finish by facing the same direction, it means that you
have completed a full revolution and also at the same time you have
swept out all the exterior angles of the polygon.
You can practise this by walking around the perimeter of the following
pentagon.
start and finish here
Walk along this circuit, sweeping out its exterior angles with your arm,
and finish at your starting point facing your starting direction.
You have rotated 360 0 , so the sum of all its exterior angles is also 360 0 .
In orienteering, if you arrive back at the same position, you have
completed a total of 360 0 worth of turning in the same direction.
The website below gives you further information about the sport of
orienteering. Perhaps you may be able to join a local club. Apart from
following compass directions, orienteering can help you read and make
maps.
Part 1
Interior and exterior angles
35
Go to the website on orienteering in Australia by visiting the CLI
webpage <http://www.cli.nsw.edu.au/Kto12>.
Select Mathematics then Stage 5.2 and follow the links to resources for
this unit SGS5 Space and geometry then select SGS5.2.1 Properties of
geometrical figures, Part 1.
Orienteering that starts and finishes at the same point always makes the
participant move through 360 0 .
You have been investigating how the sum of the exterior angles of a
polygon is related to the angle of revolution, 360 0 .
Go to the exercises section and complete Exercise 1.4 – Walking a fine
line.
36
SGS5.2.1 Properties of geometrical angles
Suggested answers – Part 1
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
a
A straight angle. A straight angle is the same as a half a turn.
It is the total of two right angles.
2
a
42 0 and x o form a straight angle and a straight angle is 180 0 .
b
138 0
x o + 42 0 = 180 0
x = 180 0 − 42 0
= 138 0
c
supplement.
20 0 is the supplement of 160 0 . Supplementary angles add
together to give 180 0 . There are many other examples, such as
this one: 10 0 is the supplement of 180 0 .
3
55 0 Remember, the angle sum of a triangle is 180 0 . This means
that the total of all three angles in any triangle is always 180 0 .
0
0
0
So ∠ABC + 56 + 69 = 180
∠ABC +125 0 = 180 0
∠ABC = 180 0 −125 0
= 55 0
4
Part 1
A regular shape has both equal angles and equal sides.
Interior and exterior angles
37
5
a
A regular octagon. An octagon has eight sides and it is regular
because all the sides and angles are equal. Note: the Greek word
‘octo’ means eight, just like in octopus, the sea creature, with
eight legs. However, don’t get confused because of the word
October, though. The month, October, use to be the eighth
month of the year with the year starting at March. This of
course also meant that December was the tenth month.
Similarly a decagon has ten sides. The beginning of this word
comes from the Greek word ‘deka’.
b
An irregular pentagon. A pentagon has five sides.
The Greek word for five is ‘pente’.
6
6 The word comes from the Greek word ‘hexa’.
7
a
i
A revolution. Remember, a revolution is a full circular turn
of 360 0 .
ii
360 0
b
A square is a regular shape, so all its angles are equal.
There are four equal angles in this revolution.
c
90 0 , revolution.
Activity – Interior angles in polygons
1
A
a
4 triangles
b
There are four triangles inside the hexagon, each with an angle
sum of 180 0 .
Angle sum of a hexagon = 4 ×180 0
= 720 0
38
SGS5.2.1 Properties of geometrical angles
c
hexagon
2
6
4 ×180 0
4
720 0
5
Angle sum of a heptagon = 7 × 180°
= 900°
A seven-sided polygon has five triangles inside it.
180°
180°
180°
180°
180°
You must multiply 180 0 by 5 not 7.
3
There are many convex octagons you could draw. Here is one
example.
A
a
6 triangles
b
i
octagon
ii
8
6
6 ×180 0
Sum of interior angles = 6 ×180 0
= 1080 0.
Part 1
Interior and exterior angles
39
iii
octagon
8
6
6 × 180°
put 1080° here
4
The number of triangles inside these polygons is always two less
than the number of sides.
5
Your polygon must have nine sides. It also must not turn in on itself,
because it must be a convex polygon. It may look like this.
This nine-sided polygon has seven triangles inside it. This is two
less than the number of sides. The angle sum of the nonagon is the
total of the 180 0 angle sums of each triangle. As there are seven of
them, this total is 7 ×180 0 .
nonagon
6
9
7 ×180 0
7
1260 0
Angle sum of a dodecagon = (12 − 2) ×180 0
= 10 ×180 0
= 1800 0
dodecagon
7
12
10
10 ×180 0
1800 0
97 ×180 0
17 460 0
Angle sum of a 99-agon = (99 − 2) ×180 0
= 97 ×180 0
= 17 460 0
99-agon
40
99
97
SGS5.2.1 Properties of geometrical angles
8
Interior angle sum for an n-agon = (n − 2) ×180 0 . You must use
brackets because all of the n − 2 is multiplied by 180 0 .
n−2
n
n-agon
(n − 2) ×180 0
(n − 2) ×180 0
Note that the last two entries are the same, because algebraic
expressions always show you how to do a calculation. You can only
get a numerical answer when you substitute a number.
9
decagon
10
8
8 ×180 0
1440 0
The completed table is shown below:
Name of
polygon
Number of
sides
Number of
triangles
inside
polygon
Calculation for
total of interior
angles.
Angle sum of
polygon
triangle
3
1
1×180 0
180 0
quadrilateral
4
2
2 ×180 0
360 0
pentagon
5
3
3 ×180 0
540 0
hexagon
6
4
4 ×180 0
720 0
heptagon
7
5
5 ×180 0
900 0
octagon
8
6
6 ×180 0
1080 0
nonagon
9
7
7 ×180 0
1260 0
decagon
10
8
8 ×180 0
1440 0
dodecagon
12
10
10 ×180 0
1800 0
99-agon
99
97
97 ×180 0
17 460 0
n-agon
n
n−2
(n − 2) ×180 0
(n − 2) ×180 0
Activity – Regular polygons
1
Part 1
a
a regular octagon. You could also say a regular convex octagon.
Regular means that the angles are equal and octagon means it
has eight angles. Convex means that all the angles at the
vertices are less than 180 0 .
Interior and exterior angles
41
b
eight
c
x°
C
There are six triangles inside this octagon.
d
i
1080 0
Angle sum of octagon = 6 ×180 0
= 1080 0
ii
8 There are eight equal angles and the total of these angles
is 1080 0
iii
x o = 135 0
x o = 1080 0 ÷ 8
= 135 0
Activity – Exterior angles of a polygon
1
a
A straight angle is 180 0 . x o and 30 0 form the straight angle.
x o is the supplement of 30 0 .
b
x o = 150 0
30 0 + x o = 180 0
x o = 180 0 − 30 0
= 150 0
50°
30° 150°
100°
42
SGS5.2.1 Properties of geometrical angles
c
The exterior angle next to 100 0 is 80 0 and the exterior angle
next to 50 0 is 130 0 .
Each exterior angle is the supplement of the interior angle next
to it. If you subtract the interior angle from the straight angle of
180 0 you will get the correct answer. Like this:
180° – 50°
130°
50°
180° – 100°
d
2
30° 150°
100°
80°
i
130 0 + 80 0 +150 0 = 360 0
ii
The sum of the exterior angles of a triangle is 360 0
a
180° – 35°
145°
60°
35°
180° – 110°
b
120°
95°
85°
110°
70°
180° – 95°
i
60 0 + 85 0 + 70 0 +145 0 = 360 0
ii
The sum of the exterior angles of a quadrilateral is 360 0
iii The sums of 360 0 are both the same.
3
Part 1
a
120 0 +150 0 +100 0 +110 0 + 60 0 = 540 0
Interior and exterior angles
43
b
You can draw your exterior angles clockwise or anticlockwise.
120°
120°
60°
60° 120°
150°
60°
70°
110°
30°
150°
60°
30°
120°
100°
110°
80°
100°
80°
70°
All these exterior angles are calculated in the same way. The
interior and exterior angle form a straight angle, so to calculate
the exterior angle you must subtract the interior angle from 180 0
180 0 −120 0 = 60 0
180 0 −150 0 = 30 0
180 0 −100 0 = 80 0
180 0 −110 0 = 70 0
180 0 − 60 0 = 120 0
c
i
360 0 ( 60 0 + 30 0 + 80 0 + 70 0 +120 0 = 360 0 )
ii
The sum (or total) of the exterior angles of a pentagon is
360 0
iii The sum of 360 0 , for the exterior angles of all three
polygons, is the same.
4
d
The sum of the exterior angles of any convex polygon is always.
360 0 .
a
1080 0 Notice below, that when you make n = 8, the word
polygon changes to octagon.
Interior angle sum of an octagon = (n − 2) ×180 0
= (8 − 2) ×180 0
= 6 ×180 0
= 1080 0
b
135 0
Each interior angle = 1080 0 ÷ 8
= 135 0
44
SGS5.2.1 Properties of geometrical angles
c
45 0
Each exterior angle = 180 0 −135 0
= 45 0
If each interior angle is 135 0 , then each exterior angle is the
supplement of 135 0 , because they form a straight angle.
This means that:
135°
d
exterior angles = 45°
135°
Yes
Sum of exterior angles of regular octagon = 8 × 45 0
= 360 0
Activity – Walking a fine line
1
180 0 ( 90 0 + 90 0 = 180 0 ) Your should have your back to the object.
2
270 0 ( 180 0 + 90 0 = 270 0 ) You have rotated three quarters of the
way around a circle.
3
a
You are facing the object again. You have rotated a full circle.
b
360 0 . It doesn’t matter where you start. So long as you finish
in the same direction, you must have swept through 360 0 .
4
No. You have only walked forward. You have not turned yet.
5
c
You have only turned a quarter turn and your arm has swept
around the outside of the rectangle, not the inside. The angle
must be a right angle and it must be outside the rectangle on
your left as you turn.
Part 1
Interior and exterior angles
45
6
The angle below is drawn with a little square in the corner. This
indicates that the angle is 90 0
7
8
46
SGS5.2.1 Properties of geometrical angles
9
10 No. The object should be to your left.
11
12 a
i
90 0 . They are all right angles.
ii
360 0 . ( 90 0 + 90 0 + 90 0 + 90 0 = 360 0 )
b
Yes. They are exterior angles.
c
Yes they are.
d
exterior
All four angles are the exterior angles of the rectangle.
Part 1
Interior and exterior angles
47
e
360 0 , sum, polygon, revolution.
The total amount of turning while walking around this rectangle
is 360 0 . This is the same as the sum of the exterior angles of a
convex polygon and the same as a full rotation which is called a
revolution.
48
SGS5.2.1 Properties of geometrical angles
Exercises – Part 1
Exercises 1.1 to 1.4
Name
___________________________
Teacher
___________________________
Exercise 1.1 – Interior angles in polygons
1
The formula, Angle sum of a polygon = (n − 2) ×180 0 is used to
calculate the angle sum of a polygon that has n sides (or n angles.)
a
Write in your own words the meaning of “angle sum of a
polygon”.
___________________________________________________
___________________________________________________
b
If a polygon has 24 sides, what is the value of n that you must
use in order to calculate its angle sum? ____________________
c
Use this value of n to calculate the angle sum of a polygon with
24 angles.
___________________________________________________
___________________________________________________
Part 1
Interior and exterior angles
49
2
Answer the following questions about the polygon below.
83°
95°
x°
55°
a
What is the name of this polygon?________________________
b
Complete: Any polygon with four sides has an angle sum of
_________.
c
Calculate the value of x o .
___________________________________________________
___________________________________________________
3
Answer the following questions about the polygon below.
95°
x°
130°
110°
A
120°
a
What is the name of this polygon?________________________
b
On the diagram above, draw diagonals from vertex A.
i
How many triangles are there inside this polygon? _______
ii
The angle sum of this shape is 540 o . Show this is true by
doing the correct working below.
________________________________________________
________________________________________________
50
SGS5.2.1 Properties of geometrical angles
c
Calculate the value of x o .
___________________________________________________
___________________________________________________
4
The sum of eleven of the angles of a dodecagon is 1750 0
a
Use the formula: Angle sum of a polygon = (n − 2) ×180 0 to
calculate the total of all the angles inside the dodecagon.
___________________________________________________
___________________________________________________
b
Calculate the size of the remaining angle.
___________________________________________________
___________________________________________________
5
The angle sum of a polygon is 2880 0 . How many sides does it
have?
_______________________________________________________
_______________________________________________________
6
It doesn’t matter whether a polygon is convex or non-convex.
The sum of the interior angles is the same number.
Answer the following questions to show that this is true for a
non-convex octagon.
a
There are two triangles already drawn
for you inside the non-convex octagon.
Carefully divide the rest of the shape
into four more triangles. (Do not draw
lines that cross over each other.)
b
Show that the formula:
Angle sum of a polygon = (n − 2) ×180 0
180°
180°
is true for this octagon.
___________________________________________________
___________________________________________________
Part 1
Interior and exterior angles
51
Exercise 1.2 – Regular polygons
1
a
What is the sum of all the interior angles of a hexagon?
___________________________________________________
b
All the angles of a regular hexagon are the same. (Its sides also
have the same length.)
What is the size of each angle in a regular hexagon?
___________________________________________________
c
Complete the row marked ‘regular hexagon’ in the table below
by writing your answers in the correct place.
d
The table below lists some regular polygons. It contains the
angle sum and size of every angle in each of these shapes.
Finish the table below. (Use the lines below the table to do
your working.)
Name of regular
polygon
Interior angle sum
Size of every angle
Equilateral triangle
180 0
180 0 ÷ 3 = 60 0
1080 0
1080 0 ÷ 8 = 135 0
Square
Regular pentagon
Regular hexagon
Regular decagon
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
52
SGS5.2.1 Properties of geometrical angles
2
Calculate the size of one of the angles in a regular, 20-sided figure.
_______________________________________________________
_______________________________________________________
3
Find the value of x o , giving reasons.
A
B
x°
C
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
4
The diagram below shows three tessellated regular hexagons.
x°
a
Calculate the size of x o . Show your working below.
___________________________________________________
___________________________________________________
Part 1
Interior and exterior angles
53
b
Write a sentence to explain why you chose to calculate x 0 , the
way you did above.
___________________________________________________
___________________________________________________
c
There is another way you can find x o . Show your solution
below, giving reasons.
___________________________________________________
___________________________________________________
d
Complete the sentence below:
Each angle of a regular hexagon is equal to__________.
A regular ___________________ tessellates because a
revolution of 360 0 is divisible by ___________.
e
Explain why a regular pentagon does not tessellate.
___________________________________________________
___________________________________________________
___________________________________________________
54
SGS5.2.1 Properties of geometrical angles
Exercise 1.3 – Exterior angles of a polygon
1
The exterior angles of an equilateral triangle are each 120 0 .
These angles can be found by doing the following calculation.
Interior angle = 180 0 ÷ 3 (Angle sum of an equilateral triangle)
= 60 0
Exterior angle = 180 0 − 60 0 (A straight angle)
= 120 0
120°
60°
60°
120°
a
60° 120°
Do a different calculation to find 120 0 by using the rule, ‘The
sum of the exterior angles of a polygon is 360 0 . Put this rule, in
brackets, next to your calculation.
___________________________________________________
___________________________________________________
b
Calculate and then complete the table for each of the regular
polygons below.
Regular polygon
Size of exterior angle
Equilateral triangle
120 0
Square
90 0
Regular pentagon
Regular hexagon
Regular octagon
Regular decagon
Part 1
Interior and exterior angles
55
2
Calculate the size of x o in the following polygons giving reasons.
79°
25°
x°
52°
37°
a
___________________________________________________
___________________________________________________
x°
b
___________________________________________________
___________________________________________________
3
Use a pencil, a blank piece of paper, a fifty-cent coin, a ruler and a
protractor to do the following task.
Carefully trace your fifty-cent coin onto a piece of paper using a
sharp pencil.
Use a ruler to draw one exterior angle. (Make sure this line goes in a
straight line ( 180 0 ) from one side of the dodecagon.)
Make the line quite long.
Use a ruler to extend the other side of this angle. Make it quite long.
Measure this angle with a protractor. Write your answer here. _____
56
SGS5.2.1 Properties of geometrical angles
How close is your practical measurement with the theoretical
measurement in the question above? __________________________
4
a
Calculate the size of the exterior angle of these polygons.
i
A regular 100-sided polygon. ________________________
________________________________________________
ii
A regular 1000-sided polygon ________________________
________________________________________________
ii
What happens to the size of the exterior angle as the number
of sides of the polygon increases? _____________________
________________________________________________
5
The exterior angles of a regular polygon are each 20 0 .
How many vertices does this polygon have?
_______________________________________________________
Part 1
Interior and exterior angles
57
Exercise 1.4 – Walking a fine line
1
2
Identify on the diagrams below, the angles you sweep through as you
walk around this polygon in:
a
an anticlockwise direction
b
a clockwise direction.
Explain in your own words why the sum of the exterior angles of a
convex polygon is the same as the amount of turning done during a
walk around its boundary. Use a diagram if you wish.
_______________________________________________________
_______________________________________________________
_______________________________________________________
58
SGS5.2.1 Properties of geometrical angles
3
The interior angles of this polygon are shown below.
a
The sum of these interior angles is 900 0 . Explain why this is
correct for all seven-sided polygons.
___________________________________________________
___________________________________________________
b
Imagine you are starting from point A and walking
anticlockwise along the boundary of this polygon. Draw all the
exterior angles you must sweep through so that you arrive back
at your exact starting position.
i
Calculate the size of all of these exterior angles.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
ii
Show that the sum of these exterior angles is 360 0 .
________________________________________________
________________________________________________
Part 1
Interior and exterior angles
59