Lines and angles

© Boardworks Ltd 2004 1 of 69

KS3 Mathematics

S1 Lines and Angles


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SS1.4

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Contents

S1 Lines and angles

S1.1 Labelling lines and angles

S1.3 Calculating angles

S1.2 Parallel and perpendicular lines

S1.4 Calculating angles in triangles andquadrilaterals


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 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.


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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.

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.


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.


Parallel or perpendicular?


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AS1.3 Calculating angles



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°


Getting to know angles

Use SMILE programs

Angle 90 and Angle 360

To get to know angles.


You must learn facts about angles.So you can calculate their size without drawing or measuring.

• Learn facts about

• Angles between intersecting lines

• Angles on a straight line

• Angles around a point


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.Vertically opposite angles are equal.


Angles on a straight line


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.


Angles around a point


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.


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°


You can use the facts you have learnt to calculate angles.

Work out the answers to the following ten ticks questions.


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.


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.


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 equalCorresponding 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 equalAlternate angles are equal

ab

ce

g

h


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


Calculating angles

Calculate the size of angle a.

a29º

46º

Hint: Add another line.

a = 29º + 46º = 75º


Calculating angles involving parallel lines.

Calculate these angles from this ten ticks worksheet.


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S1.4 Angles in triangles and quadrilaterals




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°.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°


Calculating angles in a triangle.

Calculate the angles shown on this ten ticks worksheet.


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.


Angles in an isosceles triangle

For example,

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°


Calculating angles in special triangles.

Calculate the angles on this ten ticks worksheet.


Interior angles in triangles

c a

b

The angles inside a triangle 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°.


Exterior angles in triangles

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.

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°.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 °


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,

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°

Technology

Lines and angles