KS3 Mathematics

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KS3 Mathematics

S1 Lines and Angles


S1.1 Labelling lines and angles

Contents

S1 Lines and angles

S1.4 Angles in polygons

S1.3 Calculating angles

S1.2 Parallel and perpendicular lines


Lines

In Mathematics, a straight line is defined as having infinite length and no width.

Is this possible in real life?


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.


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 to the other.

We label angles using capital letters.

A

BC

This angle can be described as ABC or ABC or B.


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.

For example, an equilateral triangle has three equal sides and three equal angles.

A derived property follows from a definition.

For example, the angles in an equilateral triangle are each 60°.

60°60°

60°


Convention, definition or derived property?



Contents




S1 Lines and angles


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.


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.


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.

Where do you see parallel lines in everyday life?

This means that they are always equidistant.


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.

ab

cd Each angle is 90. We show

this with a small square in each corner.


Parallel or perpendicular?


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.


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.


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.



Contents



S1 Lines and angles



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°


Angles


A half turn measures 180°.

This is a straight line.180°


Angles


A three-quarter turn measures 270°.

270°


Angles


A full turn measures 360°.360°


Intersecting lines


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.


Angles on a straight line


Angles on a straight line

Angles on a line add up to 180.

a + b = 180°

ab

because there are 180° in a half turn.


Angles around a point


Angles around a point

Angles around a point add up to 360.

a + b + c + d = 360

a b

cd

because there are 360 in a full turn.


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°


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.


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.


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


Angles made with parallel lines


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 equal


ee

aab

c

d

f

g

h



b

c

d

f

g

h

a = e because



gg

cc



c = g because


ab d

ef h


ff



b = f because


ab

c

d

e

g

h

b


ff

dd

Alternate angles

There are two pairs of alternate angles, or Z-angles.

d = f because

Alternate angles are equal

ab

ce

g

h


ccee

Alternate angles

There are two pairs of alternate angles, or Z-angles.

c = e because

Alternate angles are equal

ab

g

h

d

f


Calculating angles

Calculate the size of angle a.

a29º

46º

Hint: Add another line.

a = 29º + 46º = 75º



Contents



S1 Lines and angles



Angles in a triangle



For any triangle,

a b

c

a + b + c = 180°

The angles in a triangle add up to 180°.



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°.


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°


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.


Angles in an isosceles triangle

For example,

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°

a

a

88° + 2a = 180°2a = 92°

a = 46°

46°

46°


Polygons

A polygon is a 2-D shape made when line segments enclose a region.

A

B

C D

EThe line segments are called sides.

The end points are called vertices. One of these is called a vertex.

2-D stands for two-dimensional. These two dimensions are length and width. A polygon has no height.


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


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°.


Exterior angles in polygons

f

d

e

When we extend the sides of a polygon outside the shape

exterior angles are formed.


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.

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


Interior and exterior angles in a triangle


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°


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


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°.


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 °


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) = 560°

The sum of the interior angles in a pentagon is 560°.

c

a

b

and g + h + i = 180°

d

f

egih


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.


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°.


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°


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°.


Interior and exterior angles in a square

In a square,





90° 90°

90° 90°

90°

90°

90°

90°


Interior and exterior angles in a regular pentagon

In a regular pentagon,





108°

108° 108°

108° 108°

72°72°

72°

72°

72°


Interior and exterior angles in a regular hexagon

In a regular hexagon,





120° 120°

120° 120°

120° 120°

60°

60°

60°

60°

60°

60°


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°.


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°.


Take Turtle for a walk


Find the number of sides


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 = 140º = 150º

KS3 Mathematics

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