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.

Although most of the geometry we look at 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.



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.




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?

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.



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.

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 AB BC.



Line segments AB and CD, however, are parallel. In mathematics this is written as AB CD, 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.



Click on the figure below to interact with the model.

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.



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.



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.



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.