Types of Line
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.
Lines in planes
Although most of the geometry we look at is two dimensional (flat), we sometimes extend it to
solid
Three-dimensional shapes are often referred to as 'solids'.
solid
objects. It is therefore good to have some understanding of three dimensional space and its elements. Have a look at the animation below.

 Figure 1. Building a 3-D space.

When two lines exist within a single
plane
A plane has position, length and width but no height. It is an object with two-dimensions.
plane
, we say that they are 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.

 Figure 2. Coplanar and non-coplanar lines.
In Fig.2, which line or lines are
coplanar
Any set of points, lines, curves and/or shapes are coplanar if they exist in the same plane.
coplanar
with line 1?
• Looking at the animation, no plane that includes line 1 also includes line 2, so line 2 is not coplanar with line 1.

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?

 Figure 3. Quantum Cloud, by Anthony Gormley.
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 line segment is the set of points on the straight line between any two points, including the two endpoints themselves.
line segment
'.

A section of a line between and including any two points on that line is a line
segment
The region of a circle bounded by an arc and the chord joining its two end points.
segment
. For instance, look at the figure below.

 Figure 4. The line through A and B.

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.

 Figure 5. Lines and line segments.
Select the
point
• A point has no properties except position. It is an object with zero dimensions.
• Points in the x-y plane can be specified using x and y coordinates.
point
in Fig.5 that fits each description.
•  Lies on line AD, line CB and line segment CB U V W X Z Lies on line CD but not on line segment CD U V W X Z Lies on line AB and on line segment AB U V W X Z Lies on line BC but not on line segment BC U V W X Z Lies on line BC and on line AD U V W X Z Lies on line AC, line segment AC and line BD U V W X Z
Parallel and perpendicular lines
When two coplanar lines (or line segments) have an
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
of 90° between them, we say that they are perpendicular to one another. So we can say of the tennis court pictured below that AB is
perpendicular
Two lines or planes are perpendicular if they are at right angles to one another.
perpendicular
to BC. Sometimes the symbol is used to mean 'is perpendicular to', so ABBC.

 Figure 6. Perpendicular lines.

Line segments AB and CD, however, are parallel. In mathematics this is written as ABCD, where the symbol means 'is
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
parallel
to'. Like all parallel lines, line segments AB and CD are
equidistant
Items that are at an equal distance from an identified point, line or plane are said to be equidistant from it.
equidistant
and the lines of which they are part never meet (or '
intersect
To intersect is to have a common point or points. For example, two lines intersect at a point and two planes intersect at a straight line. The point at which two or more lines intersect is called a vertex.
intersect
'), even though they are in the same plane. The rails of a railway track, for instance, are parallel.

Consider three coplanar lines: EF, GH and KL. If EF is perpendicular to GH, and KL is also perpendicular to GH, which two of the following could be true (though not at the same time)? (You may find it helpful to manipulate the lines in the model below.)
• No! These lines may be parallel but they cannot be perpendicular
• Well done!
• No! Draw a diagram to see why.

Click on the figure below to interact with the model.

 Figure 7.  Three coplanar lines.

In Fig.8, the two planes are parallel to one another. Are the two lines also parallel to one another?
 Figure 8. Two planes, each containing one line.
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 intersects two or more coplanar lines.
transversal
.

 Figure 9. Transversals.

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
Opposite angles are formed when two lines intersect.
Angle 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.

 Figure 10. Angles created by a transversal.

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
Corresponding angles are created when a transversal or line segment intersects two other lines or line segments.
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.

 Figure 11. Angles created by a transversal.

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.

 Figure 12. Angles created by a transversal.

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

• c and e

• d and f

 Figure 13. Angles created by a transversal.
Complete the following paragraph with reference to Fig.13, using each angle only once.

• The vertically opposite angle to b is and the vertically opposite angle to e is . Line 1 is the transversal which means that d and are alternate (or 'Z') angles. Another set of alternate angles are and g. In total there are 4 pairs of corresponding angles. These are: g and ; d and ; h and ; c and .
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.
Exercises

1. Select the correct words to complete the paragraph below.

• A is simply a position in space and has neither , width nor height. A is a set of points and has but no or height. A consists of a set of lines and has both length and . A set of planes is a , which has length, width and .
2. Match each picture with the correct description. They are all either lines or line segments.
 Figure 14. Look for relationships between these lines.
3. Select the correct relationship between the pairs of lines in Fig.14
•  lines a and b no relation parallel perpendicular lines a and d no relation parallel perpendicular lines a and f no relation parallel perpendicular lines b and c no relation parallel perpendicular lines b and e no relation parallel perpendicular lines b and f no relation parallel perpendicular lines c and d no relation parallel perpendicular lines c and g no relation parallel perpendicular lines d and f no relation parallel perpendicular lines e and f no relation parallel perpendicular
4. Which of the line segments in Fig.15 below are transversals?
 Figure 15. Transversal lines and angles.
5. Complete the table below with the missing values of the angles in Fig.15 above. The figure is also shown below in Fig.16.

•  angle value a 80 b c d 100 e f g 108 h i j 124 k l

 Figure 16. Transversal lines and angles.
6. Use the angles in Fig.16 above to describe the different pairs of angles as either vertically opposite, corresponding ('F') or alternate ('Z') angles.
•  Angles a and c alternate (Z) angles corresponding (F) angles vertically opposite angles Angles b and d alternate (Z) angles corresponding (F) angles vertically opposite angles Angles a and e alternate (Z) angles corresponding (F) angles vertically opposite angles Angles b and h alternate (Z) angles corresponding (F) angles vertically opposite angles Angles h and f alternate (Z) angles corresponding (F) angles vertically opposite angles Angles d and l alternate (Z) angles corresponding (F) angles vertically opposite angles Angles c and i alternate (Z) angles corresponding (F) angles vertically opposite angles Angles b and j alternate (Z) angles corresponding (F) angles vertically opposite angles Angles b and l alternate (Z) angles corresponding (F) angles vertically opposite angles
Well done!
Try again!