© boardworks ltd 2004 1 of 69 algebra a lines and angles

© Boardworks Ltd 2004 1 of 69

Algebra A

Lines and Angles


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 CBA.


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.


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°


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


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 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 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°


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


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


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°.


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 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°


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°.

© boardworks ltd 2004 1 of 69 algebra a lines and angles

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