Introduction

Most of us are quite good at recognizing whether the shapes that we see around us are straight or crooked, upright, or squint. In this way every one of us could be described as a natural geometer.

Sometimes, though, it's necessary to be quite specific in the language we use and the measurements we make. In this unit we learn how to talk about, understand and compare lines.

Most of us are quite good at recognizing whether the shapes that we see around us are straight or crooked, upright, or squint. In this way every one of us could be described as a natural geometer.

Sometimes, though, it's necessary to be quite specific in the language we use and the measurements we make. In this unit we learn how to talk about, understand and compare lines.

Lines in planes

Although most of the geometry we look at is two dimensional (flat), we sometimes extend it to

**solid**When two lines exist within a single

**plane****coplanar**. Click on Fig.2 to see a plane (in the form of a grid) appear. Clicking and dragging on the grid will rotate the plane about the red line. All points that lie in this plane are coloured red.

A British artist, Anthony Gormley, used the idea of lines intersecting in a three dimensional space to produce the 29-metre tall sculpture pictured below. By varying how densely the 'lines' are packed into the space, he managed to make a human form appear in the midst of the structure. Can you see it?

Lines and line segments

In mathematics, lines are infinite – that is, they have no ends. It is not often, in the real world, that we deal with such
lines. When people use the word 'line' to talk about the lines of a tennis court, for example, what they actually mean is
'

**line segment**A section of a line between and including any two points on that line is a line

**segment**An infinite line has been drawn through points

*A*and

*B*. Let us call this line 'line 1'. Now press the button in Fig.4 above to highlight the line segment

*AB*. It includes the points

*A*and

*B*and all the points on line 1 between

*A*and

*B*. (We say that line segment

*AB*can be

**produced**to form line 1.)

The four points

*A*,

*B*,

*C*and

*D*in Fig.5 below are coplanar (they all exist in one plane). The line segments between these four points are coloured red. Use this picture to answer the question below.

Parallel and perpendicular lines

When two coplanar lines (or line segments) have an

**angle****perpendicular**to one another. So we can say of the tennis court pictured below that

*AB*is

**perpendicular***BC*. Sometimes the symbol is used to mean 'is perpendicular to', so

*AB*

*BC*.

Line segments

*AB*and

*CD*, however, are

**parallel**. In mathematics this is written as

*AB*

*CD*, where the symbol means 'is

**parallel***AB*and

*CD*are

**equidistant**

**intersect**Click on the figure below to interact with the model.

Transversal lines

Any line that intersects two or more coplanar lines is called a **transversal**. In each of the examples below,

*t*is a

**transversal**A transversal line creates several interesting angle pairs.

**Vertically opposite angles**

Any two angles that are opposite one another at a vertex (intersection) are called vertically

**opposite angles***a*is opposite angle

*c*and angle

*b*is opposite angle

*d*. Opposite angles are always equal.opposite angles. Press the button in Fig.10 below to show two such angles,

*a*and

*c*. These angles are always equal. Vertically opposite angles are created when any two lines intersect.

So in Fig.10 above we can see these four pairs of vertically opposite angles:

*a*and*c**b*and*d**e*and*g**f*and*h*

**Corresponding angles**

Angles

*c*and

*g*in Fig.11 below form a pair of

**corresponding angles***a*and

*e*are corresponding angles. If the two intersected lines are parallel, then the corresponding angles are equal. They are also known as 'F' angles.corresponding angles. Press the button to highlight these angles and the 'F' shape that they make.

So there are four corresponding angle pairs:

*a*and*e**b*and*f**c*and*g**d*and*h*

**Alternate angles**

Alternate angle pairs are those like

*d*and

*f*in Fig.12 below. They make a 'Z' shape, as can be seen by pressing the button.

There are only two alternate angle pairs in Fig.12 and these are:

*c*and*e**d*and*f*

Summary

Lines are composed of a set of points.

The set of points between and including any two points on a line is called a line segment.

If two coplanar lines never intersect, they are parallel to one another.

Two lines that intersect at an angle of 90° are perpendicular to one another.

A line that intersects at least two others is called a transversal.

A transversal line creates pairs of opposite, corresponding ('F'), and alternate ('Z') angles.

Lines are composed of a set of points.

The set of points between and including any two points on a line is called a line segment.

If two coplanar lines never intersect, they are parallel to one another.

Two lines that intersect at an angle of 90° are perpendicular to one another.

A line that intersects at least two others is called a transversal.

A transversal line creates pairs of opposite, corresponding ('F'), and alternate ('Z') angles.

Well done!

Try again!