2
Angles
CHAPTER
2.1 Triangles
b
b
c
Fit angles a, b and c
together. They make a
straight line.
Tear off its corners.
Draw a triangle on paper and
label its angles a, b and c.
c
a
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°
180° 129° 51°
x 51°
72°
57°
Add 72° and 57°
Take the result away from 180°, as the angle sum of a triangle is 180°.
State the size of angle x.
Sometimes the fact that the angle sum of a triangle is 180° and other angle facts are needed.
Example 2
Work out the size of
b angle y.
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°.
60°
b 180° 53° 127°
y 127°
67°
x
y
Sum of angles on a straight line is 180°.
17
Angles
CHAPTER 2
Exercise 2A
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
a
37°
3
39°
66°
57°
119°
b
c
4
5
d
32°
6
e
43°
27°
28°
7
64°
8
9
76°
k
j
30°
h i
f
l
60°
34°
36°
g
48°
10
11
12
81°
v
59°
119° m
n
57°
q
p
t
r 123°
s
u
41°
114°
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°
2.2 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°
18
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.
2.2 Equilateral triangles and isosceles triangles
CHAPTER 2
Example 3
y
Work out the size of
a angle x
Give reasons for your answers.
Solution 3
a x 41°
b
b angle y.
x
41°
Isosceles triangle with equal angles opposite equal sides.
41° 41° 82°
180° 82° 98°
Angle sum of triangle is 180°.
y 98°
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°
Isosceles triangle with equal angles opposite equal sides.
x 17°
Exercise 2B
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
3
b
4
58°
e
29°
a
69°
5
116°
c
6
7
d
8
58°
h
73°
m
i
g
k
l
f
j
9
10
11
u
q
p
n 106°
s
r 80°
12
v
t
62°
128°
19
Angles
CHAPTER 2
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
2.3 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
Fit angles a, b, c and d
together at a point.
Tear off its corners.
d
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
98°
Work out the size of angle x.
118°
76°
Solution 5
76° 118° 98° 292°
360° 292° 68°
x 68°
20
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.
2.3 Quadrilaterals
CHAPTER 2
Example 6
75°
a Write down the size of angle x.
b Work out the size of angle y.
Give a reason for each answer.
x
72°
y
121°
Solution 6
a x 75°
Where two straight lines cross, the opposite angles are equal.
b 121° 72° 75° 268°
360° 268° 92°
Angle sum of a quadrilateral is 360°
y 92°
Example 7
109°
The diagram shows a kite.
a Write down the size of angle x.
Give a reason for each answer.
b Work out the size of angle y.
y
83°
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°
Angle sum of a quadrilateral is 360°.
x
Exercise 2C
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
83°
76°
5
2
98°
113°
b
64°
a
71°
6
e
3
58°
4
67°
c
109°
124°
121°
74°
7
115°
d
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°
21
Angles
CHAPTER 2
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°
w
47°
37°
119°
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°
e
116°
19
g 68°
69°
80°
18
74°
h
113° i
20
m
n
56°
38°
l
2.4 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.
22
j
134°
f
143°
61°
k
2.4 Polygons
CHAPTER 2
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
10 180 1800
The sum of the angles 1800°
Subtract 2 from the number of sides to find the number of triangles.
Multiply the number of triangles by 180.
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.
23
Angles
CHAPTER 2
Example 9
Find the size of each interior angle of a regular decagon.
Solution 9
10 2 8
8 180 1440
1440 10 144
Each interior angle is 144°
Subtract 2 from the number of sides to find the number of triangles.
Multiply the number of triangles by 180 to find the sum of all 10 interior angles.
All 10 interior angles are the same size. So divide 1440 by 10
State the size of each interior angle.
Example 10
The diagram shows a regular 9-sided polygon
(nonagon) with centre O.
O
a Work out the size of
i angle x ii angle y.
x
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°
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 2D
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.
3 A polygon can be split into 17 triangles by drawing diagonals from one corner.
How many sides has the polygon?
24
2.4 Polygons
CHAPTER 2
In Questions 4–9, find the size of each of the angles marked with letters and show your working.
4
5
6
109°
97°
c
117°
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°
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.
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.
25
Angles
CHAPTER 2
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
21
b angle i.
The diagram shows a hexagon.
Work out the size of
a
angle j
124°
116°
93°
127°
83°
h
b angle k.
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.
23 The diagram shows a quadrilateral.
n
a Work out the size of each of the angles marked
with letters.
m
106°
94°
b Work out l m n p
102° l
58°
p
24 The diagram shows a pentagon.
s
a Work out the size of each of the angles marked
with letters.
t
b Work out q r s t u
117°
r
81°
145°
124° q
73°
u
25 The diagram shows a hexagon.
a Work out the size of each of the angles marked
with letters.
y 123°
x
129°
w
152°
b Work out u v w x y z
z
85°
163°
68° v
u
2.5 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 angle exterior angle 180°
The sum of the exterior angles of any polygon is 360°.
26
interior
angle
exterior angle
2.5 Exterior angles
CHAPTER 2
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 facing in
d
P
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,
b the size of each interior angle.
Solution 12
a 360° 18 20°
Because the polygon is regular, all 18 exterior angles are equal.
Their sum is 360° and so divide 360° by 18
b 180° 20° 160°
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
27
Angles
CHAPTER 2
Exercise 2E
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.
2.6 Corresponding angles and alternate angles
Parallel lines are always the same distance apart. They never meet.
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.
28
2.6 Corresponding angles and alternate angles
CHAPTER 2
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.
Notice that they form an F shape.
b Angle s is the alternate angle to the shaded angle.
Notice that they form a Z shape.
Example 15
a
b
Find the size of angle x.
Give a reason for your answer.
78° x
Solution 15
a x 78°
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.
67°
p
q
Solution 16
a 180° 67° 113°
p 113°
b The sum of the angles on a straight line is 180°.
c q 113°
d Corresponding angles.
29
Angles
CHAPTER 2
Exercise 2F
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
30
b
53°
j
52°
75°
154° a
61°
d
i
h
g
e
62°
f
2.7 Proofs
CHAPTER 2
18
19
20
l
k
m
78°
59°
37°
64°
2.7 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
The diagram shows a triangle PQR.
R
T
b
Extend the side PQ to S.
At Q draw a line QT parallel to PR.
Then
and
angle x angle a (corresponding angles)
angle y angle b (alternate angles)
P
y
a
x
Q
S
xyab
Adding,
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.
Proof 2
The angle sum of a triangle is 180°
This proof starts in the same way as Proof 1.
R
T
b
The diagram shows a triangle PQR.
Extend the side PQ to S.
At Q draw a line QT parallel to PR.
Then
angle x angle a (corresponding angles)
and
angle y angle b (alternate angles)
Adding,
P
a
c
y
x
Q
S
xyab
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.
Proof 3
The opposite angles of a parallelogram are equal
Draw a diagonal of the parallelogram.
c
angle a angle c (alternate angles)
angle b angle d (alternate angles)
b
d
a
Adding, a b c d which proves that the statement is true.
31
Angles
CHAPTER 2
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.
b Give a reason for your answer.
c Find the size of angle y.
d Give reasons for your answer.
Solution 18
a x 71°
b Opposite angles of a parallelogram are equal.
2 71° 142°
c
y
71°
d Angle sum of a quadrilateral is 360°.
360° 142° 218°
Opposite angles of a parallelogram are equal.
218° 2 109°
y 109°
Exercise 2G
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°
2.8 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.
32
n
2.8 Bearings
CHAPTER 2
Example 19
Example 20
Measure the bearing
of B from A.
Measure the bearing of
Q from P.
N
N
Q
B
Solution 19
From North, measure
the angle clockwise.
The angle is 52°.
So the bearing
is 052°.
Solution 20
To find the angle clockwise
from North with a semi-circular
protractor measure the shaded
anticlockwise angle (38°)
and subtract it from 360°.
A
N
N
360° 38° 322°
B
P
The bearing of Q from
P is 322°.
Q
N
A
Q
P
P
Example 21
Folkestone and Dover are shown on the map.
N
The bearing of a ship from Folkestone is 117°.
N
The bearing of the ship from Dover is 209°.
Draw an accurate diagram to show the position
of the ship.
Dover
Folkstone
Mark the position with a cross X. Label it S.
Solution 21
Draw a line on a bearing of 117° from Folkestone.
N
Draw a line on a bearing of 209° from Dover by
measuring an angle of 151° (360° 209°)
anticlockwise from North.
N
Dover
209°
Folkstone 117°
(Alternatively, measure an angle of 29° clockwise
from South.)
S
Put a X where the lines cross.
Label the position S.
33
Angles
CHAPTER 2
Sometimes, answers to questions have to be worked out, not found using a protractor.
Example 22
N
The bearing of B from A is 061°.
Work out the bearing of A from B.
Diagram NOT
accurately drawn
N
B
61°
Solution 22
The bearing of A from B is the reflex angle at B.
A
N
y 61° (alternate angles)
Diagram NOT
accurately drawn
N
Bearing 180° 61°
B
y
241°
61°
A
Exercise 11H
In Questions 1–4, measure the bearing of Q from P.
1
2
N
N
Q
Q
P
3
P
4
N
N
P
P
Q
Q
34
2.8 Bearings
5
6
7
8
9
CHAPTER 2
Draw diagrams similar to those in Questions 1–4 to show the bearings
a 026°
b 217°
c 109°
d
The diagram shows two points, A and B.
The bearing of a point L from A is 048°.
The bearing of L from B is 292°.
On the diagram on the resource sheet find the
position of L by making an accurate drawing.
The diagram shows two points, P and Q.
The bearing of a point M from P is 114°.
The bearing of M from Q is 213°.
On the diagram on the resource sheet find the
position of M by making an accurate drawing.
N
N
A
B
N
N
P
Q
Cromer and Great Yarmouth are shown on the map.
The bearing of a ship from Cromer is 052°.
The bearing of the ship from Great Yarmouth is 348°.
On the diagram on the resource sheet find the position
of the ship by making an accurate drawing.
Mark the position of the ship with a cross X.
Label it S.
The bearing of Q from P is 038°.
Work out the bearing of P from Q.
10
334°.
N
N
Cromer
Great
Yarmouth
The bearing of T from S is 146°.
Work out the bearing of S from T.
N
N
Diagram NOT
accurately drawn
N
N
38°
P
11
Q
S
146°
Diagram NOT
accurately drawn
T
The bearing of B from A is 074°.
The bearing of C from B is 180°.
AB AC.
Work out the bearing of C from A.
N
N
Diagram NOT
accurately drawn
B
74°
A
C
12
The diagram shows the positions of
York, Scarborough and Hull.
The bearing of Scarborough from
York is 052°.
The bearing of Hull from York is 118°.
The distance between York and Scarborough is
the same as the distance between York and Hull.
Work out the bearing of Hull from Scarborough.
Scarborough
N
52°
Diagram NOT
accurately drawn
York
Hull
35
Angles
CHAPTER 2
Chapter summary
You should know and be able to use these facts
The angle sum of a triangle is 180°.
An equilateral triangle has three equal
angles and three equal sides.
60°
60°
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.
A quadrilateral is a shape with four straight sides and four angles.
The angle sum of a quadrilateral is 360°.
A polygon is a shape with three or more straight sides.
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.
The angle sum of a polygon can be found by subtracting 2 from the number
of sides and multiplying the result by 180°.
A polygon with all its sides the same length and all its angles
the same size is called a regular polygon.
At a vertex, interior angle exterior angle 180°.
The sum of the exterior angles of any polygon is 360°.
To find the size of each exterior angle of a regular
polygon, divide 360° by the number of sides.
Where a straight line crosses two parallel lines,
the corresponding angles are equal.
Where a straight line crosses two parallel lines,
the alternate angles are equal.
Bearings are measured clockwise from North.
interior
angle
You should also know these proofs
36
An exterior angle of a triangle is equal to the sum of the interior angles
at the other two vertices.
The angle sum of a triangle is 180°.
exterior angle
Chapter 2 review questions
CHAPTER 2
Chapter 2 review questions
In Questions 1–12, find the size of each of the angles marked with a letter.
The diagrams are not accurately drawn.
1
2
71°
3
4
c
56°
64°
f
a
e
94° b
58°
38°
d
5
6
h
7
i
109°
8
k
64°
118° g
9
m
94°
84°
10
92°
j
114°
117°
11
128°
12
64°
57°
l
141°
112°
53°
n
r
113°
q p
t
s
13 In triangle ABC, AB AC and angle C 50°.
a Write down the special name of triangle ABC.
b Work out the value of y.
A
Diagram NOT
accurately drawn
y°
B
50°
C
(1385 June 1999)
14 Calculate the value of x.
x°
150°
Diagram NOT
accurately drawn
25°
45°
(4400 November 2004)
15 Work out the value of a.
78°
102°
Diagram NOT
accurately drawn
63°
a°
43°
(1388 March 2002)
16 Work out the size of each exterior angle of a regular 10-sided polygon.
17 a Work out the sum of the interior angles of a 9-sided polygon.
The size of each exterior angle of a regular polygon is 20°.
b Work out how many sides the polygon has.
37
Angles
CHAPTER 2
18 The diagram shows a regular hexagon.
a Work out the value of x.
b Work out the value of y.
Diagram NOT
accurately drawn
y°
x°
(1385 June 2001)
19 ABCDE is a regular pentagon.
AEF and CDF are straight lines.
Work out the size of angle DFE.
Give reasons for your answer.
A
E
F
20 a
b
(1388 March 2004)
A
55°
B
D
21 AC BC
AB is parallel to DC
Angle ABC 52°
a Work out the value of
C
D
i Write down the size of the
angle marked x.
ii Give a reason for your answer.
i Write down the size of the
angle marked y.
ii Give a reason for your answer.
x
y
Diagram NOT
accurately drawn
C
75°
E
(1384 November 1996)
A
D
Diagram NOT
accurately drawn
p°
i p
Diagram NOT
accurately drawn
B
ii q
The angles marked p° and r ° are equal.
b What geometrical name is given to
this type of equal angles?
r° q°
52°
C
B
(1384 November 1997)
22 The diagram represents the positions
of Wigan and Manchester.
a Measure and write down the
bearing of Manchester
from Wigan.
b Find the bearing of Wigan
from Manchester.
N
N
Wigan
(1385 June 1998)
Manchester
23 Measure the bearing of A from B.
N
B
A
38
(1388 March 2004)
Chapter 2 review questions
CHAPTER 2
24 Work out the bearing of
ii B from P,
ii P from A.
N
A
63°
138°
P
Diagram NOT
accurately drawn
(1387 November 2004)
B
25 The diagram shows the position of each of three buildings in a town.
N
N
Hospital
Diagram NOT
accurately drawn
Cinema
72°
Art
gallery
The bearing of the Hospital from the Art gallery is 072°.
The Cinema is due east of the Hospital.
The distance from the Hospital to the Art gallery is equal to the distance from the Hospital to
the Cinema.
Work out the bearing of the Cinema from the Art gallery.
(1387 November 2003)
39