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© Boardworks Ltd 2006 1 of 69
KS3 Mathematics
S1 Lines and Angles
© Boardworks Ltd 2006 2 of 69
A
A
A
A
Contents
S1 Lines and angles
S1.1 Labelling lines and angles
S1.4 Angles in polygons
S1.3 Calculating angles
S1.2 Parallel and perpendicular lines
© Boardworks Ltd 2006 3 of 69
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.
The line segment shown below has end points A and B.
A B
We can call this line ‘line segment AB’.
© Boardworks Ltd 2006 5 of 69
Labelling angles
When two lines meet at a point an angle is formed.
An angle is a measure of the rotation of one of the line segments relative to the other.
We label points using capital letters.
A
BC
The angle can then be described as ABC or ABC or B.
Sometimes instead an angle is labelled with a lower case letter.
© Boardworks Ltd 2006 6 of 69
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.
The statement that an equilateral triangle has three equal sides and three equal angles is a definition.
A derived property follows from a definition.
For example, the angles in an equilateral triangle are each 60°.
60°60°
60°
© Boardworks Ltd 2006 7 of 69
Convention, definition or derived property?
© Boardworks Ltd 2006 8 of 69
Contents
A
A
A
A
S1.2 Parallel and perpendicular lines
S1.4 Angles in polygons
S1.1 Labelling lines and angles
S1.3 Calculating angles
S1 Lines and angles
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Lines in a plane
What can you say about these pairs of lines?
These lines cross, or intersect.
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.
Parallel lines will never meet. They stay an equal distance apart.
Where do you see parallel lines in everyday life?
We can say that parallel lines are always equidistant.
© Boardworks Ltd 2006 12 of 69
Perpendicular lines
What is special about the angles at the point of intersection here?
a = b = c = d
Lines that intersect at right angles are called perpendicular lines.
Lines that intersect at right angles are called perpendicular lines.
ab
cd Each angle is 90. We show
this with a small square in each corner.
© Boardworks Ltd 2006 13 of 69
Parallel or perpendicular?
© Boardworks Ltd 2006 14 of 69
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.
The shortest distance from a point to a line is always the perpendicular distance.
© Boardworks Ltd 2006 15 of 69
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.
© Boardworks Ltd 2006 16 of 69
Drawing parallel lines with a set square
We can also draw parallel lines using a ruler and a set square.
Slide the set square along the ruler to draw a line parallel to the first.
Place the set square on the ruler and use it to draw a straight line perpendicular to the ruler’s edge.
© Boardworks Ltd 2006 17 of 69
Contents
A
A
A
AS1.3 Calculating angles
S1.4 Angles in polygons
S1.1 Labelling lines and angles
S1.2 Parallel and perpendicular lines
S1 Lines and angles
© Boardworks Ltd 2006 18 of 69
Angles
Angles are measured in degrees.
A quarter turn measures 90°.
It is called a right angle.
We label a right angle with a small square.
90°
© Boardworks Ltd 2006 19 of 69
Angles
Angles are measured in degrees.
A half turn measures 180°.
This is a straight line.180°
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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.
A full turn measures 360°.360°
© Boardworks Ltd 2006 22 of 69
Intersecting lines
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Vertically opposite angles
When two lines intersect, two pairs of vertically opposite angles are formed.
a
b
c
d
a = c and b = d
Vertically opposite angles are equal.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.Angles on a line add up to 180.
a + b = 180°
ab
because there are 180° in a half turn.
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Angles around a point
© Boardworks Ltd 2006 27 of 69
Angles around a point
Angles around a point add up to 360.Angles around a point add up to 360.
a + b + c + d = 360
a b
cd
because there are 360 in a full turn.
© Boardworks Ltd 2006 28 of 69
b c
d
43° 43°
68°
Calculating angles around a point
Use geometrical reasoning to find the size of the labelled angles.
103°
a167°
137°
69°
© Boardworks Ltd 2006 29 of 69
Complementary angles
When two angles add up to 90° they are called complementary angles.
When two angles add up to 90° they are called complementary angles.
ab
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.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.
Which angles are equal to each other?
ab
c
d
ef
g
h
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Angles made with parallel lines
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dd
hh
ab
ce
f
g
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
ab
ce
f
g
d = h because
Corresponding angles are equalCorresponding angles are equal
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ee
aab
c
d
f
g
h
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b
c
d
f
g
h
a = e because
Corresponding angles are equalCorresponding angles are equal
© Boardworks Ltd 2006 35 of 69
gg
cc
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
c = g because
ab d
ef h
Corresponding angles are equalCorresponding angles are equal
© Boardworks Ltd 2006 36 of 69
ff
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
b = f because
ab
c
d
e
g
h
b
Corresponding angles are equalCorresponding angles are equal
© Boardworks Ltd 2006 37 of 69
ff
dd
Alternate angles
There are two pairs of alternate angles, or Z-angles.
d = f because
Alternate angles are equalAlternate angles are equal
ab
ce
g
h
© Boardworks Ltd 2006 38 of 69
ccee
Alternate angles
There are two pairs of alternate angles, or Z-angles.
c = e because
ab
g
h
d
f
Alternate angles are equalAlternate angles are equal
© Boardworks Ltd 2006 39 of 69
Calculating angles
Calculate the size of angle a.
a29º
46º
Hint: Add another line.
a = 29º + 46º = 75º
© Boardworks Ltd 2006 40 of 69
Contents
A
A
A
A
S1.4 Angles in polygons
S1.1 Labelling lines and angles
S1.3 Calculating angles
S1.2 Parallel and perpendicular lines
S1 Lines and angles
© Boardworks Ltd 2006 41 of 69
Angles in a triangle
© Boardworks Ltd 2006 42 of 69
Angles in a triangle
For any triangle:
a b
c
a + b + c = 180°
The angles in a triangle add up to 180°.The angles in a triangle add up to 180°.
© Boardworks Ltd 2006 43 of 69
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.
These angles are equal because they are alternate angles.
a
a
b
b
Call this angle c.
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°.
© Boardworks Ltd 2006 44 of 69
Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles.
233°
82°31°
116°
326°
43°49°
28°
ab
c
d
33°64°
88°
25°
© Boardworks Ltd 2006 45 of 69
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 of 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
Find the sizes 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°
a
a
88° + 2a = 180°2a = 92°
a = 46°
46°
46°
© Boardworks Ltd 2006 47 of 69
Polygons
A polygon is a 2-D shape made when line segments enclose a region.
A polygon is a 2-D shape made when line segments enclose a region.
A
B
C D
EThe line segments are called sides.
Each end point is called a vertex. We call more than one vertices.
2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no thickness.
© Boardworks Ltd 2006 48 of 69
Number of sides Name of polygon
3
4
5
6
7
8
9
10
Naming polygons
Polygons are named according to the number of sides they have.
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
© Boardworks Ltd 2006 49 of 69
Interior angles in polygons
c a
b
The angles inside a polygon are called interior angles.
The sum of the interior angles of a triangle is 180°.The sum of the interior angles of a triangle is 180°.
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Exterior angles in polygons
f
d
e
When we extend the sides of a polygon outside the shape
exterior angles are formed.
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Interior and exterior angles in a triangle
ab
c
Any exterior angle in a triangle is equal to the sum of the two opposite interior angles.
Any exterior angle in a triangle is equal to the sum of the two opposite interior angles.
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.
bc
© Boardworks Ltd 2006 52 of 69
Interior and exterior angles in a triangle
© Boardworks Ltd 2006 53 of 69
Calculating angles
Calculate the size of the lettered angles in each of the following triangles.
82°31°64° 34°
ab
33°116°
152°d25°
127°
131°
c
272°
43°
© Boardworks Ltd 2006 54 of 69
Calculating angles
Calculate the size of the lettered angles in this diagram.
56°a
73°b86° 69°
104°
Base angles in the isosceles triangle = (180º – 104º) ÷ 2= 76º ÷ 2= 38º
38º 38º
Angle a = 180º – 56º – 38º = 86ºAngle b = 180º – 73º – 38º = 69º
© Boardworks Ltd 2006 55 of 69
Sum of the interior angles in a quadrilateral
c
ab
What is the sum of the interior angles in a quadrilateral?
We can work this out by dividing the quadrilateral into two triangles.
d f
e
a + b + c = 180° and d + e + f = 180°
So: (a + b + c) + (d + e + f ) = 360°
The sum of the interior angles in a quadrilateral is 360°.The sum of the interior angles in a quadrilateral is 360°.
© Boardworks Ltd 2006 56 of 69
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 °
Do you know the sum of the interior angles for any other polygons?
a b
c
We have just shown that the sum of the interior angles in any quadrilateral is 360°.
a
bc
d
a + b + c + d = 360 °
© Boardworks Ltd 2006 57 of 69
Sum of the interior angles in a pentagon
What is the sum of the interior angles in a pentagon?
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°
So: (a + b + c) + (d + e + f ) + (g + h + i) = 540°
The sum of the interior angles in a pentagon is 540°.The sum of the interior angles in a pentagon is 540°.
c
a
b
and g + h + i = 180°
d
f
egih
© Boardworks Ltd 2006 58 of 69
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.
How many triangles can a hexagon be divided into?A hexagon can be divided into four triangles.
© Boardworks Ltd 2006 59 of 69
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.
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°.
The sum of the interior angles in an n-sided polygon is (n – 2) × 180°.
© Boardworks Ltd 2006 60 of 69
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
Size of each interior angle
Equilateral triangle 180° 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°
© Boardworks Ltd 2006 61 of 69
Interior and exterior angles in an equilateral triangle
In an equilateral triangle:
60°
60°
Every interior angle measures 60°.
Every exterior angle measures 120°.
120°
120°
60°120°
The sum of the interior angles is 3 × 60° = 180°.
The sum of the exterior angles is 3 × 120° = 360°.
© Boardworks Ltd 2006 62 of 69
Interior and exterior angles in a square
In a square:
Every interior angle measures 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°.
90° 90°
90° 90°
90°
90°
90°
90°
© Boardworks Ltd 2006 63 of 69
Interior and exterior angles in a regular pentagon
In a regular pentagon:
Every interior angle measures 108°.
Every exterior angle measures 72°.
The sum of the interior angles is 5 × 108° = 540°.
The sum of the exterior angles is 5 × 72° = 360°.
108°
108° 108°
108° 108°
72°72°
72°
72°
72°
© Boardworks Ltd 2006 64 of 69
Interior and exterior angles in a regular hexagon
In a regular hexagon:
Every interior angle measures 120°.
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°.
120° 120°
120° 120°
120° 120°
60°
60°
60°
60°
60°
60°
© Boardworks Ltd 2006 65 of 69
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°.
© Boardworks Ltd 2006 66 of 69
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°.The sum of the exterior angles in a polygon is 360°.
© Boardworks Ltd 2006 67 of 69
Take Turtle for a walk
© Boardworks Ltd 2006 68 of 69
Find the number of sides
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Calculate the missing angles
50º
This pattern has been made with three different shaped tiles.
The length of each side is the same.
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 = 40º = 130º = 140º = 130º = 140º