11
Angles 2
CHAPTER
11.1 Triangles
Draw a triangle on paper
and label its angles a, b
and c.
b
b
c
a
Fit angles a, b and c
together. They make a
straight line.
Tear off its corners.
c
a
a
b
c
This shows that the angles in this triangle add up to 180° but it is not a proof.
That comes later in this chapter.
The angles on a straight line add up to 180° and so the angles in this triangle add up to 180°.
The angle sum of a triangle is 180°.
Example 1
x
Work out the size of angle x.
Solution 1
72° 57° 129°
Add 72° and 57°
180° 129° 51°
Take the result away from 180°, as the angle sum of a triangle is 180°.
x 51°
72°
57°
State the size of angle x.
Sometimes both the fact that the angle sum of a triangle is 180° and the angle facts from
Chapter 3 are needed.
Example 2
Work out the size of
a angle x
Give reasons for your answers.
Solution 2
a 67° 60° 127°
180° 127° 53°
x 53°
Angle sum of triangle is 180°.
b angle y.
b 180° 53° 127°
y 127°
60°
67°
x
y
Sum of angles on a straight line is 180°.
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Angles 2
CHAPTER 11
Exercise 11A
In this exercise, the triangles are not accurately drawn.
In Questions 1–12, find the size of each of the angles marked with letters and show your
working.
77°
121°
1
2
3
4
d
a
22°
39°
33°
32°
27°
57°
37°
66°
119°
b
c
5
88°
6
e
28°
64°
f=61° g=119°7
h=110° i=70°
8
43°
34°
g
f
76°
36°
i=90°
60°
j
30°
h i
k=75° l=75°
9
10
l
11
k
119° m
48°
12
81°
n
t=114° u=41°
v=25°
v
59°
q
p
s
57°
m=61° n=60° p=120°
t
r 123°
u
41°
114°
q=81° r=57° s=42°
In Questions 13–15, find the size of each of the angles marked with letters and show your
working.
Give reasons for your answers.
13
14
15
y
84°
z
48°
24°
137° w
x
42°
58°
11.2 Equilateral triangles and isosceles triangles
An equilateral triangle
has three equal sides and
three equal angles.
As the angle sum of a
triangle is 180°, the size
of each angle is
180 3 60°.
60°
60°
190
60°
An isosceles triangle has
two equal sides and the
angles opposite the equal
sides are equal.
A triangle whose sides
are all different lengths
is called a scalene
triangle.
11.2 Equilateral triangles and isosceles triangles
CHAPTER 11
Example 3
Work out the size of
a angle x
Give reasons for your answers.
Solution 3
a x 41°
b 41° 41° 82°
180° 82° 98°
y 98°
y
b angle y.
x
41°
Isosceles triangle with equal angles opposite equal sides.
Angle sum of triangle is 180°.
Example 4
x
Work out the size of angle x.
Give reasons for your answer.
Solution 4
180° 146° 34°
146°
Angle sum of triangle is 180°.
34° 2° 17°
x 17°
Isosceles triangle with equal angles opposite equal sides.
Exercise 11B
In this exercise, the triangles are not accurately drawn.
In Questions 1–12, find the size of each of the angles marked with letters and show your
working.
1
2
b
3
4
58°
e
29°
a
69°
5
116°
c
6
7
h
d
8
58°
73°
m
i
g
k
l
f
j
9
10
11
u
q
p
n 106°
s
r 80°
12
v
t
62°
128°
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Angles 2
CHAPTER 11
In Questions 13–15, find the size of each of the angles marked with letters and show
your working.
Give reasons for your answers.
13
14
104°
15
x
w
68°
42°
y
11.3 Quadrilaterals
A quadrilateral is a
shape with four straight
sides and four angles.
To find the angle sum of
a quadrilateral, draw a
quadrilateral on paper
and label its angles
a, b, c and d.
d
a
Tear off its corners.
d
Fit angles a, b, c and d
together at a point.
c
b a
c
d
a
b
c
b
The angles at a point add up to 360° and so this shows that the angles in this quadrilateral
add up to 360°.
The angle sum of a quadrilateral is 360°.
To prove this result, draw a diagonal of the quadrilateral.
The diagonal splits the quadrilateral into two triangles.
The angle sum of each triangle is 180°.
So the angle sum of the quadrilateral is 2 180° 360°.
Example 5
Work out the size of angle x.
98°
118°
76°
Solution 5
76° 118° 98° 292°
360° 292° 68°
x 68°
192
x
Add 76, 118 and 98
Take the result away from 360, as the angle sum of a quadrilateral is 360°.
State the size of angle x.
11.3 Quadrilaterals
CHAPTER 11
Example 6
75°
a Write down the size of angle x.
b Work out the size of angle y.
x
Give reasons for your answer.
72°
y
121°
Solution 6
a x 75°
b 121° 72° 75° 268°
360° 268° 92°
y 92°
Where two straight lines cross, the opposite angles are equal.
Angle sum of a quadrilateral is 360°
Example 7
109°
The diagram shows a kite.
a Write down the size of angle x.
b Work out the size of angle y.
y
83°
Give reasons for your answer.
x
Solution 7
a x 109°
b 83° 109° 109° 301°
360° 301° 59°
y 59°
A kite has a line of symmetry.
Angle x is a
reflection of
the 109° angle
and so the two
angles are equal.
109°
y
83°
x
Angle sum of a quadrilateral is 360°.
Exercise 11C
In this exercise, the quadrilaterals are not accurately drawn.
In Questions 1–12, find the size of each of the angles marked with letters and show your
working.
1
2
3 67°
4
d
83°
113°
b
98°
76°
5
c
64°
a
71°
6
e
58°
109°
124°
121°
74°
7
115°
118°
8
147°
66°
48°
g
f
j
143° i
96°
112°
h
9
58°
82°
k
116°
l
10
4
83°
94°
n
m 41°
11
75°
126° p
12
u
r
q
s
84°
113°
t
129°
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Angles 2
CHAPTER 11
13 The diagram shows a kite.
a Write down the size of angle v.
b Work out the size of angle w.
14 The diagram shows a kite.
Work out the value of x.
x
126°
47°
w
119°
37°
v
x
15 The diagram shows an isosceles trapezium.
a Write down the value of a.
b Work out the value of b.
b
b
62°
a
In Questions 16–20, find the sizes of the angles
marked with letters and show your working.
Give reasons for your answers.
16
17
55°
h
74°
69°
g 68°
e
80°
116°
19
61°
k
113° i
j
134°
f
20
m
n
56°
143°
18
38°
l
11.4 Polygons
A polygon is a shape with three or more straight sides.
Some polygons have special names.
A 3-sided polygon is called a triangle.
A 4-sided polygon is called a quadrilateral.
A 5-sided polygon is called a pentagon.
A 6-sided polygon is called a hexagon.
An 8-sided polygon is called an octagon.
A 10-sided polygon is called a decagon.
To find the sum of the angles of a polygon, split it into triangles.
For example, for this hexagon, draw as many diagonals as possible from one corner.
This splits the hexagon into four triangles.
The angle sum of a triangle is 180° and so the
sum of the angles of a hexagon is 4 180° 720°.
Sometimes, these angles are called interior angles to
emphasise that they are inside the polygon.
194
11.4 Polygons
CHAPTER 11
Using this method, the sum of the interior angles of any polygon can be found.
Number of sides Number of triangles Sum of the interior angles
4
2
360°
5
3
540°
6
4
720°
7
5
900°
8
6
1080°
9
7
1260°
10
8
1440°
The number of triangles into which the polygon can be split up is always two less than the
number of sides.
Example 8
Find the sum of the angles of a 12-sided polygon (dodecagon).
Solution 8
12 2 10
Subtract 2 from the number of sides to find the number of triangles.
10 180 1800
Multiply the number of triangles by 180.
The sum of the angles 1800°
State the sum of the angles in degrees.
A polygon with all its sides the same length and all its angles the same size is called a
regular polygon.
So a square is a regular polygon, because all its sides are
the same length and all its angles are 90°, but a rhombus
is not a regular polygon.
Although its sides are all the same length, its angles are not all the same size.
Here are three more regular polygons.
a regular pentagon
a regular hexagon
a regular octagon
The Pentagon in Washington DC
is the headquarters of the
US Department of Defence.
Bees’ honeycomb is made
up of regular hexagons.
Regular octagons tessellate
with squares.
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Angles 2
CHAPTER 11
Example 9
Find the size of each interior angle of a regular decagon.
Solution 9
10 2 8
Subtract 2 from the number of sides to find the number of triangles.
8 180 1440
Multiply the number of triangles by 180
to find the sum of all 10 interior angles.
1440 10 144
All 10 interior angles are the same size. So divide 1440 by 10
Each interior angle is 144°
State the size of each interior angle.
Example 10
The diagram shows a regular 9-sided polygon
(nonagon) with centre O.
a Work out the size of i angle x ii angle y.
b Use your answer to part a ii to work out the
size of each interior angle of the polygon.
Solution 10
a i x 360° 9
x 40°
O
x
y
Each corner of the polygon could be joined to the centre
O to make 9 equal angles at O. The total of all 9 angles
is 360°, as altogether they make a complete turn.
State the size of angle x.
(40° is the angle at the centre
of a regular 9-sided polygon.)
ii 180° 40° 140
140° 2° 70°
y 70°
b 2 70° 140°
Each interior angle is 140°.
The angle sum of a triangle is 180° and so
the sum of the two base angles is 140°.
The triangle is isosceles and so
the two base angles are equal.
State the size of angle y.
Because the polygon is regular, it has nine lines
of symmetry and each interior angle is twice
the size of each base angle of the triangle.
State the size of each interior angle.
Exercise 11D
In this exercise, the polygons are not accurately drawn.
1 Find the sum of the angles of a 15-sided polygon.
2 Find the sum of the angles of a 20-sided polygon.
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11.4 Polygons
CHAPTER 11
3 A polygon can be split into 17 triangles by drawing diagonals from one corner.
How many sides has the polygon?
In Questions 4–9, find the size of each of the angles marked with letters and show your
working.
4
5
6
109°
97°
104°
88°
121°
82°
94°
81°
7
b
a
126°
147°
8
128°
132°
121°
118°
9
136°
129°
124°
162°
118°
153°
e
104°
131°
c
117°
123°
114°
134°
122°
140°
d
f
10 The diagram shows a pentagon.
All its sides are the same length.
a Work out the value of g.
b Is the pentagon a regular polygon?
Explain your answer.
137°
60°
g
g
11 Work out the size of each interior angle of
a a regular pentagon
b a regular hexagon
c
a regular octagon.
12 Work out the size of each interior angle of a regular 15-sided polygon.
13 Work out the size of each interior angle of a regular 20-sided polygon.
14 Work out the size of the angle at the centre of a regular pentagon.
15 Work out the size of the angle at the centre
of a regular 12-sided polygon.
Australia’s 50 cent
coin is a regular
12-sided polygon
(dodecagon)
16 The angle at the centre of a regular polygon is 60°.
How many sides has the polygon?
17 The angle at the centre of a regular polygon is 20°.
a How many sides has the polygon?
b Work out the size of each interior angle of the polygon.
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Angles 2
CHAPTER 11
18 a Work out the angle at the centre of a regular octagon.
b Draw a circle with a radius of 5 cm and, using your answer to part a , draw a
regular octagon inside the circle.
19 a Work out the angle at the centre of a regular 10-sided polygon.
b Draw a circle with a radius of 5 cm and, using your answer to part a , draw a
regular 10-sided polygon inside the circle.
20 The diagram shows a pentagon.
Work out the size of
a angle h b angle i.
21 The diagram shows a hexagon.
Work out the size of
a angle j b angle k.
116°
124°
93°
127°
83°
h
i
142°
97°
87°
68° j
k
22 Craig says, ‘The sum of the interior angles of this polygon is 1000°’.
Explain why he must be wrong.
n
23 The diagram shows a quadrilateral.
a Work out the size of each of the angles marked
with letters.
b Work out l m n p
94°
124° l
58°
p
s
24 The diagram shows a pentagon.
a Work out the size of each of the angles marked
with letters.
b Work out q r s t u
25 The diagram shows a hexagon.
a Work out the size of each of the angles marked
with letters.
b Work out u v w x y z
m
106°
t
117°
r
81°
145°
124° q
73°
u
y 123°
x
129°
w
152°
z
85°
163°
68° v
u
11.5 Exterior angles
A polygon’s interior angles are the angles inside the polygon.
Extend a side to make an exterior angle, which is outside the polygon.
At each vertex (corner), the interior angle and the exterior
angle are on a straight line and so their sum is 180°.
interior
interior angle exterior angle 180°
angle
The sum of the exterior angles of any polygon is 360°.
198
exterior angle
11.5 Exterior angles
CHAPTER 11
R
To show this, imagine someone standing at P on this
c
b
quadrilateral, facing in the direction of the arrow.
Q
They turn through angle a, so that they are facing in
the direction PQ, and then walk to Q.
a
S
At Q, they turn through angle b, so that they are
d
P
facing in the direction QR, and then walk to R.
At R, they turn through angle c, so that they are facing in the direction RS, and then walk to S.
At S, they turn through angle d.
They are now facing in the direction of the arrow again and so they have turned through 360°.
The total angle they have turned through is also the sum of the exterior angles of the
quadrilateral.
So a b c d 360°
The same argument can be used with any polygon, not just a quadrilateral.
Example 11
The sizes of four of the exterior angles of a pentagon are 67°, 114°, 58° and 73°.
Work out the size of the other exterior angle.
Solution 11
67° 114° 58° 73° 312°
Add the four given exterior angles.
360° 312° 48°
Subtract the result from 360.
Exterior angle 48°
State the size of the exterior angle.
Example 12
For a regular 18-sided polygon, work out
a the size of each exterior angle,
Solution 12
a 360° 18 20°
b 180° 20° 160°
b the size of each interior angle.
Because the polygon is regular, all 18 exterior angles
are equal. Their sum is 360° and so divide 360° by 18.
At a corner, the sum of the interior angle and the
exterior angle is 180°. So subtract 20° from 180°.
Example 13
The size of each interior angle of a regular polygon is 150°. Work out
a the size of each exterior angle,
b the number of sides the polygon has.
Solution 13
a 180° 150° 30°
b 360° 30 12
At a corner, the sum of the interior angle and the
exterior angle is 180°. So subtract 150° from 180°.
Because the polygon is regular, all the exterior angles
are 30°. Their sum is 360° and so divide 360 by 30.
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Angles 2
CHAPTER 11
Exercise 11E
1 At a vertex (corner) of a polygon, the size of the interior angle is 134°.
Work out the size of the exterior angle.
2 At a vertex of a polygon, the size of the exterior angle is 67°.
Work out the size of the interior angle.
3 The sizes of three of the exterior angles of a quadrilateral are 72°, 119° and 107°.
Work out the size of the other exterior angle.
4 The sizes of five of the exterior angles of a hexagon are 43°, 109°, 58°, 74° and 49°.
Work out the size of the other exterior angle.
5 Work out the size of each exterior angle of a regular octagon.
6 Work out the size of each exterior angle of a regular 9-sided polygon.
7 For a regular 24-sided polygon, work out
a the size of each exterior angle,
b the size of each interior angle.
8 For a regular 40-sided polygon, work out
a the size of each exterior angle,
b the size of each interior angle.
9 The size of each interior angle of a regular polygon is 168°. Work out
a the size of each exterior angle,
b the number of sides the polygon has.
10 The size of each interior angle of a regular polygon is 170°.
Work out the number of sides the polygon has.
11.6 Corresponding angles and alternate angles
Parallel lines are always the same distance apart. They never meet. (Section 3.5)
In diagrams, arrows are used to show that lines are parallel.
In the diagram, a straight line crosses two parallel
lines. The shaded angles are called corresponding
angles and are equal to each other.
The F shape formed by
corresponding angles can be
helpful in recognising them.
Other pairs of corresponding angles have been shaded in the diagrams below.
200
11.6 Corresponding angles and alternate angles
CHAPTER 11
In the diagram, a straight line crosses two parallel lines.
The shaded angles are called alternate angles
and are equal to each other.
The Z shape formed by alternate angles
can be helpful in recognising them.
Another pair of alternate angles has been
shaded in this diagram.
Example 14
Write down the letter of the angle which is
a corresponding to the shaded angle,
b alternate to the shaded angle.
s p
r q
Solution 14
a Angle q is the corresponding angle to
the shaded angle.
b Angle s is the alternate angle to the shaded angle.
Notice that they form an F shape.
Notice that they form a Z shape.
Example 15
a Find the size of angle x.
b Give a reason for your answer.
Solution 15
a x 78°
78° x
b Alternate angles.
Example 16
a
b
c
d
Find the size of angle p.
Give a reason for your answer.
Find the size of angle q.
Give a reason for your answer.
Solution 16
a 180° 67° 113°
p 113°
c q 113°
67°
p
q
b The sum of the angles on a
straight line is 180°.
d Corresponding angles.
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Angles 2
CHAPTER 11
Exercise 11F
In this exercise, the diagrams are not accurately drawn.
1 Write down the letter of the angle which is
a corresponding to the shaded angle,
b alternate to the shaded angle.
x
u
2 Write down the letter of the angle which is
a corresponding to the shaded angle,
b alternate to the shaded angle.
y
w
v
x
w
v
In Questions 3–5, find the sizes of the angles marked with letters
and state whether the pairs of angles are corresponding or alternate.
3
4
5
75°
96°
b
a
c
118°
In Questions 6–20, find the sizes of the angles marked with letters.
Give reasons for your answers.
6
7
8
132°
g
d
f
i
e
h
82°
9
126°
10
11
79°
k
52°
m
l
n
p
j
67°
76°
q
12
r
13
t 59°
14
76°
x
v
42°
s
w
47°
82°
u
y
z
15
16
17
69°
c
202
b
53°
j
52°
75°
154° a
61°
d
i
h
g
e
62°
f
11.7 Proofs
CHAPTER 11
18
19
20
l
k
m
78°
59°
37°
64°
11.7 Proofs
In mathematics, a proof is a reasoned argument to show that a statement is always true.
The proofs which follow make use of corresponding and alternate angles.
Proof 1
An exterior angle of a triangle is equal to the sum of the interior
angles at the other two vertices
R
The diagram shows a triangle PQR.
b
Extend the side PQ to S.
At Q draw a line QT parallel to PR.
Then angle x angle a (corresponding angles)
a
P
and angle y angle b (alternate angles)
Adding, x y a b
x y is the exterior angle of the triangle and a b is the sum of the
interior angles at the other two vertices and so the statement is true.
T
y
x
Q
S
Proof 2
The angle sum of a triangle is 180°
This proof starts in the same way as Proof 1.
R
b
The diagram shows a triangle PQR.
Extend the side PQ to S.
At Q draw a line QT parallel to PR.
a
P
Then angle x angle a (corresponding angles)
and angle y angle b (alternate angles)
Adding, x y a b
As x, y and c are angles on a straight line, their angle sum is 180°, that is
x y c 180°
So
a b c 180° which proves that the statement is true.
T
c
y
x
Q
S
Proof 3
The opposite angles of a parallelogram are equal
Draw a diagonal of the parallelogram.
angle a angle c (alternate angles)
b
a
angle b angle d (alternate angles)
Adding, a b c d which proves that the statement is true.
c
d
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Angles 2
CHAPTER 11
Example 17
a Find the size of angle w.
63°
b Give a reason for your answer.
w
Solution 17
a 63° 44° 107°
w 107°
44°
b Exterior angle of a triangle.
(As the full reason is long, it may be shortened to this.)
Example 18
x
a Find the size of angle x.
c Find the size of angle y.
b Give a reason for your answer.
d Give reasons for your answer.
Solution 18
a x 71°
c
2 71° 142°
360° 142° 218°
218° 2 109°
y 109°
y
71°
b Opposite angles of a parallelogram are equal.
d Angle sum of a quadrilateral is 360°.
Opposite angles of a parallelogram are equal.
Exercise 11G
In this exercise, the diagrams are not accurately drawn.
Find the size of each of the angles marked with letters.
Give reasons for your answers.
1
2
3
4
c
62°
d
38°
a
47°
b
5
6
e
64°
f
71°
78°
g
h
7
j
i
118°
137°
104°
8
k
127°
m
39° l
117°
9
10
p
42°
36°
q
r
47°
11.8 Bearings
Bearings are used to describe directions.
Bearings are measured clockwise
from North.
When the angle is less than 100°, one or two zeros are written in front of the angle, so
that the bearing still has three figures.
204
n