These occur when we have 2 lines that NEVER cross each other. For this to happen the two lines need to have exactly same slope. If they have different slopes they will cross.

Parallel lines occur often in the real world.

Have a look at this wacky picture! Are the horizontal lines parallel?

In fact they are, this is a visual illusion. The spacing of the black rectangles makes our eyes think otherwise.

Perpendicular

The Leaning Tower of Pisa

Perpendicular is the word used to describe when one object meets another at exactly 90°. So perpendicular lines are simply lines that cross each other at exactly 90°.

To see how important the idea of perpendicular really is just think about your floor, walls and roof. If a builder does not take care to make the walls perpendicular to the floor and ceiling you'll end up with an unstable house.

The leaning tower of Pisa is a famous example of perpendicular angles gone wrong! Prior to restoration work performed between 1990 and 2001, the tower leaned at an angle of 5.5°, but the tower now leans at about 3.99°. That means the acute angle made by the tower and the ground is 86.01°.

The following applet will allow you to see parallel and perpendicular lines in action. Watch this video if you would like to see this interactive in action then have a go yourself

Created with Geogebra

Collinear Points

When points all lie on the same line, they are called collinear points.

We can think of points being collinear in space (a) , or on the cartesian plane (b)

(a)

(b)

Intersections and concurrent lines

Because lines extend forever in both directions, unless they are parallel they will intersect somewhere. On the other hand rays and line segments may, or may not intersect even when not parallel. This is because rays and line segments have end points.

Now when 3 or more lines all pass through the same point we give those lines a special name- they are called concurrent lines.

The point of intersection is called the "point of concurrency", labelled point P below.

Line Definitions

Line: passes through two points and extends forever in both directions, notated

Line segment: starts at a point and ends at a point, notated

Ray: starts at a fixed point $A$A and extends forever through a second point $B$B, notated

Perpendicular: two lines, segments or rays are perpendicular if they cross or meet at right angles, notated

Parallel: two lines, segments or rays are parallel if they are the same distance apart. In the case of lines this means that they never cross

Concurrent: if three or more lines, rays or segments intersect at a single point they are called concurrent.

Worked Example Videos

Question 1

Identify and classify the following as either a ray, line or a line segment:

Line
A
Ray
B
Line Segment
C

QUESTION 2

From the diagram:

A network of four intersecting lines, $AB$`AB`, $GM$`GM`, $CD$`CD`, and $EF$`EF`. On $GM$`GM` are points $G$`G`, $L$`L`, $H$`H`, $I$`I` and $M$`M`, in this order starting from the leftmost point $G$`G` to the point at the rightmost $M$`M`. On $CD$`CD` are points $C$`C`, $H$`H`, $K$`K` and $D$`D`, in this order starting from the topmost point $C$`C` to bottommost point $D$`D`. On $EF$`EF` are points $E$`E`, $I$`I`, $J$`J` and $F$`F`, in this order starting from the topmost point $E$`E` to bottommost point $F$`F`. On $AB$`AB` are points $A$`A`, $L$`L`, $K$`K`, $J$`J` and $B$`B`, in this order starting from the top leftmost point $A$`A` to point at the bottom leftmost $B$`B`. Lines $AB$`AB` and $GM$`GM` intersect at point $L$`L`. Lines $CD$`CD` intersects $GM$`GM` at point $H$`H` and $AB$`AB` at point $K$`K`. Lines $EF$`EF` intersects $GM$`GM` at point $I$`I` and $AB$`AB` at point $J$`J`. A line segment $LJ$`LJ` is formed on the line $AB$`AB` with point $K$`K` between endpoints $L$`L` and $J$`J`. Both line segments $LK$`LK` and $KJ$`KJ` are each marked with an identical single tick.

What is the midpoint of the segment $LJ$LJ?
Point D
A
Point K
B
Point H
C

Lines, Segments and Rays

Types of Lines

LINES, RAYS and SEGMENTS

Special Lines

Parallel lines