KS3 Mathematics
S1 Lines and Angles
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Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1.3 Calculating angles
S1.4 Angles in polygons
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Lines
In Mathematics, a straight line is defined as having infinite
length and no width.
Is this possible in real life?
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Labelling line segments
When a line has end points we say that it has finite length.
It is called a line segment.
We usually label the end points with capital letters.
For example, this line segment
A
B
has end points A and B.
We can call this line, line segment AB.
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Labelling angles
When two lines meet at a point an angle is formed.
A
B
C
An angle is a measure of the rotation of one of the line
segments to the other.
We label angles using capital letters.
This angle can be described as
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ABC or ABC or
B.
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Conventions, definitions and derived properties
A convention is an agreed way of describing a situation.
For example, we use dashes on lines to show that they are
the same length.
A definition is a minimum set of conditions
needed to describe something.
60°
For example, an equilateral
triangle has three equal sides
and three equal angles.
A derived property follows from a
definition.
60°
60°
For example, the angles in an equilateral triangle are each 60°.
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Convention, definition or derived property?
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Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1.3 Calculating angles
S1.4 Angles in polygons
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Lines in a plane
What can you say about these pairs of lines?
These lines cross,
or intersect.
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These lines do
not intersect.
They are parallel.
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Lines in a plane
A flat two-dimensional surface is called a plane.
Any two straight lines in a plane either intersect once …
This is called
the point of
intersection.
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Lines in a plane
… or they are parallel.
We use arrow
heads to show
that lines are
parallel.
Parallel lines will never meet. They stay an equal distance
apart.
This means that they are always equidistant.
Where do you see parallel lines in everyday life?
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Perpendicular lines
What is special about the angles at
the point of intersection here?
a=b=c=d
a
b
d
c
Each angle is 90. We show
this with a small square in
each corner.
Lines that intersect at right angles are called
perpendicular lines.
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Parallel or perpendicular?
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The distance from a point to a line
What is the shortest distance from a point to a line?
O
The shortest distance from a point to a line
is always the perpendicular distance.
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Drawing perpendicular lines with a set square
We can draw perpendicular lines using a ruler and a set
square.
Draw a straight line using a ruler.
Place the set square on the
ruler and use the right angle
to draw a line perpendicular
to this line.
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Drawing parallel lines with a set square
We can also draw parallel lines using a ruler and a set
square.
Place the set square on the ruler and use it to draw a
straight line perpendicular to the ruler’s edge.
Slide the set square
along the ruler to
draw a line parallel to
the first.
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Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1.3 Calculating angles
S1.4 Angles in polygons
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Angles
Angles are measured in degrees.
A quarter turn
measures 90°.
90°
It is called a right
angle.
We label a right
angle with a small
square.
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Angles
Angles are measured in degrees.
A half turn measures
180°.
180°
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This is a straight line.
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Angles
Angles are measured in degrees.
A three-quarter turn
measures 270°.
270°
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Angles
Angles are measured in degrees.
360°
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A full turn measures
360°.
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Intersecting lines
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Vertically opposite angles
When two lines intersect, two pairs of vertically opposite
angles are formed.
a
d
b
c
a=c
and
b=d
Vertically opposite angles are equal.
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Angles on a straight line
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Angles on a straight line
Angles on a line add up to 180.
a
b
a + b = 180°
because there are 180° in a half turn.
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Angles around a point
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Angles around a point
Angles around a point add up to 360.
a
b
c
d
a + b + c + d = 360
because there are 360 in a full turn.
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Calculating angles around a point
Use geometrical reasoning to find the size of the labelled
angles.
69°
a
167°
103°
d
68°
c
43°
b
43°
137°
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Complementary angles
When two angles add up to 90° they are called
complementary angles.
a
b
a + b = 90°
Angle a and angle b are complementary angles.
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Supplementary angles
When two angles add up to 180° they are called
supplementary angles.
a
b
a + b = 180°
Angle a and angle b are supplementary angles.
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Angles made with parallel lines
When a straight line crosses two parallel lines eight
angles are formed.
a
b
d
c
e
f
h
g
Which angles are equal to each other?
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Angles made with parallel lines
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
d = h because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
a = e because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
c = g because
Corresponding angles are equal
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Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
b = f because
Corresponding angles are equal
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Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
d = f because
Alternate angles are equal
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Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
c = e because
Alternate angles are equal
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Calculating angles
Calculate the size of angle a.
29º
a
Hint: Add
another line.
46º
a = 29º + 46º = 75º
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Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1.3 Calculating angles
S1.4 Angles in polygons
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Angles in a triangle
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Angles in a triangle
c
a
b
For any triangle,
a + b + c = 180°
The angles in a triangle add up to 180°.
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Angles in a triangle
We can prove that the sum of the angles in a triangle is
180° by drawing a line parallel to one of the sides through
the opposite vertex.
a
a
b
c
b
These angles are equal because they are alternate angles.
Call this angle c.
a + b + c = 180° because they lie on a straight line.
The angles a, b and c in the triangle also add up to 180°.
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Calculating angles in a triangle
Calculate the size of the missing angles in each of the
following triangles.
116°
a
33°
31°
b
64°
326°
82°
49°
43°
d
25°
c
88°
28°
233°
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Angles in an isosceles triangle
In an isosceles triangle, two of the sides are equal.
We indicate the equal sides by drawing dashes on them.
The two angles at the bottom on the equal sides are called
base angles.
The two base angles are also equal.
If we are told one angle in an isosceles triangle we can work
out the other two.
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Angles in an isosceles triangle
For example,
88°
a
46°
a
46°
Find the size of the
other two angles.
The two unknown angles are equal so call them both a.
We can use the fact that the angles in a triangle add up to
180° to write an equation.
88° + a + a = 180°
88° + 2a = 180°
2a = 92°
a = 46°
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Polygons
A polygon is a 2-D shape made when line segments
enclose a region.
A
The line
segments
are called
sides.
B
C
E
The end points
are called
vertices. One
of these is
called a vertex.
D
2-D stands for two-dimensional. These two dimensions
are length and width. A polygon has no height.
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Naming polygons
Polygons are named according to the number of sides they
have.
Number of sides
Name of polygon
Triangle
3
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4
5
6
7
Quadrilateral
8
9
10
Octagon
Pentagon
Hexagon
Heptagon
Nonagon
Decagon
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Interior angles in polygons
The angles inside a polygon are called interior angles.
b
c
a
The sum of the interior angles of a triangle is 180°.
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Exterior angles in polygons
When we extend the sides of a polygon outside the shape
exterior angles are formed.
e
d
f
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Interior and exterior angles in a triangle
Any exterior angle in a triangle is equal to the
sum of the two opposite interior angles.
c
ca
b
b
a=b+c
We can prove this by constructing a line parallel to this side.
These alternate angles are equal.
These corresponding angles are equal.
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Interior and exterior angles in a triangle
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Calculating angles
Calculate the size of the lettered angles in each of the
following triangles.
116°
b
33°
a
64°
82°
31°
34°
c
43°
25°
d
131°
152°
127°
272°
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Calculating angles
Calculate the size of the lettered angles in this diagram.
56°
a
86°
38º
38º
73°
b
69°
104°
Base angles in the isosceles triangle = (180º – 104º) ÷ 2
= 76º ÷ 2
= 38º
Angle a = 180º – 56º – 38º = 86º
Angle b = 180º – 73º – 38º = 69º
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Sum of the interior angles in a quadrilateral
What is the sum of the interior angles in a quadrilateral?
c d
a
f
b e
We can work this out by dividing the quadrilateral into two
triangles.
a + b + c = 180°
So,
and
d + e + f = 180°
(a + b + c) + (d + e + f ) = 360°
The sum of the interior angles in a quadrilateral is 360°.
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Sum of interior angles in a polygon
We already know that the sum of the
interior angles in any triangle is 180°.
a + b + c = 180 °
a
b
d
c
c
a
b
We have just shown that the sum of
the interior angles in any quadrilateral
is 360°.
a + b + c + d = 360 °
Do you know the sum of the interior
angles for any other polygons?
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Sum of the interior angles in a pentagon
What is the sum of the interior angles in a pentagon?
c d
a
f
b e
h
g
i
We can work this out by using lines from one vertex to divide
the pentagon into three triangles .
a + b + c = 180° and d + e + f = 180° and g + h + i = 180°
So,
(a + b + c) + (d + e + f ) + (g + h + i) = 560°
The sum of the interior angles in a pentagon is 560°.
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Sum of the interior angles in a polygon
We’ve seen that a
quadrilateral can be divided
into two triangles …
… and a pentagon can be
divided into three triangles.
many triangles
can a
AHow
hexagon
can be divided
hexagon
be divided into?
into
four triangles.
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Sum of the interior angles in a polygon
The number of triangles that a polygon can be divided
into is always two less than the number of sides.
We can say that:
A polygon with n sides can be divided into (n – 2) triangles.
The sum of the interior angles in a triangle is 180°.
So,
The sum of the interior angles in an n-sided
polygon is (n – 2) × 180°.
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Interior angles in regular polygons
A regular polygon has equal sides and equal angles.
We can work out the size of the interior angles in a regular
polygon as follows:
Name of regular
polygon
Sum of the
interior angles
Equilateral triangle 180°
Size of each
interior angle
180° ÷ 3 = 60°
Square
2 × 180° = 360°
360° ÷ 4 = 90°
Regular pentagon
3 × 180° = 540°
540° ÷ 5 = 108°
Regular hexagon
4 × 180° = 720°
720° ÷ 6 = 120°
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Interior and exterior angles in an equilateral triangle
In an equilateral triangle,
Every interior angle
measures 60°.
120°
60°
120°
60°
60°
120°
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Every exterior angle
measures 120°.
The sum of the interior
angles is 3 × 60° = 180°.
The sum of the exterior
angles is 3 × 120° = 360°.
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Interior and exterior angles in a square
In a square,
Every interior angle
measures 90°.
90°
90°
90°
90°
90°
90°
90°
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90°
Every exterior angle
measures 90°.
The sum of the interior
angles is 4 × 90° = 360°.
The sum of the exterior
angles is 4 × 90° = 360°.
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Interior and exterior angles in a regular pentagon
In a regular pentagon,
Every interior angle
measures 108°.
72°
72°
108°
Every exterior angle
measures 72°.
108°
108°
72°
72° 108° 108°
72°
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The sum of the interior
angles is 5 × 108° = 540°.
The sum of the exterior
angles is 5 × 72° = 360°.
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Interior and exterior angles in a regular hexagon
In a regular hexagon,
Every interior angle
measures 120°.
60°
60°
120° 120°
120°
120°
120° 120°
60°
60°
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60°
60°
Every exterior angle
measures 60°.
The sum of the interior
angles is 6 × 120° = 720°.
The sum of the exterior
angles is 6 × 60° = 360°.
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The sum of exterior angles in a polygon
For any polygon, the sum of the interior and exterior angles
at each vertex is 180°.
For n vertices, the sum of n interior and n exterior angles is
n × 180° or 180n°.
The sum of the interior angles is (n – 2) × 180°.
We can write this algebraically as 180(n – 2)° = 180n° – 360°.
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The sum of exterior angles in a polygon
If the sum of both the interior and the exterior angles is 180n°
and the sum of the interior angles is 180n° – 360°,
the sum of the exterior angles is the difference between
these two.
The sum of the exterior angles = 180n° – (180n° – 360°)
= 180n° – 180n° + 360°
= 360°
The sum of the exterior angles in a polygon is 360°.
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Take Turtle for a walk
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Find the number of sides
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Calculate the missing angles
This pattern has been
made with three different
shaped tiles.
The length of each side is
the same.
50º
What shape are the tiles?
Calculate the sizes of
each angle in the pattern
and use this to show that
the red tiles must be
squares.
= 50º
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= 40º
= 130º
= 140º
= 140º
= 150º
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