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KS3 Mathematics
S1 Lines and Angles
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Contents
S1 Lines and angles
A S1.1 Labelling lines and angles
A S1.2 Parallel and perpendicular lines
A S1.3 Calculating angles
SS1.4
S1.4 Calculating angles in triangles andquadrilaterals
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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 relative to the other.
We label points using capital letters.
The angle can then be described as
ABC or ABC or
B.
Sometimes instead an angle is labelled with a lower case letter.
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Contents
S1 Lines and angles
A S1.1 Labelling lines and angles
A S1.2 Parallel and perpendicular lines
A S1.3 Calculating angles
S1.4 Calculating angles in triangles andquadrilaterals
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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.
We can say that parallel lines 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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Contents
S1 Lines and angles
A S1.1 Labelling lines and angles
A S1.2 Parallel and perpendicular lines
A S1.3 Calculating angles
S1.4 Calculating angles in triangles andquadrilaterals
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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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Getting to know angles
Use SMILE programs
Angle 90 and Angle 360
To get to know angles.
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You must learn facts about angles.
So you can calculate their size without drawing or measuring.
•
•
•
•
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Learn facts about
Angles between intersecting lines
Angles on a straight line
Angles around a point
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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°
68°
d
c
43°
b
43°
137°
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You can use the facts you have learnt to calculate angles.
Work out the answers to the following ten ticks questions.
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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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Calculating angles involving parallel lines.
Calculate these angles from this
ten ticks worksheet.
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Contents
S1 Lines and angles
A S1.1 Labelling lines and angles
A S1.2 Parallel and perpendicular lines
A S1.3 Calculating angles
A S1.4 Angles in triangles and quadrilaterals
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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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Calculating angles in a triangle.
Calculate the angles shown on
this ten ticks worksheet.
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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 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
For example,
88°
a
46°
a
46°
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° + 2a = 180°
2a = 92°
a = 46°
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Calculating angles in special triangles.
Calculate the angles on this ten
ticks worksheet.
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Interior angles in triangles
The angles inside a triangle 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 triangles
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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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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