definition
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel.
Parallel Lines, and Pairs of Angles
Parallel Lines
Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember:
Always the same distance apart and never touching.
The red line is parallel to the blue line in both these cases:
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Example 1
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Example 2
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Parallel lines also point in the same direction.
Parallel lines have so much in common. It's a shame they will never meet!
http://www.mathsisfun.com/geometry/parallel-lines.html
This is easy! Look at these parallel lines:
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What do they have in common?
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So, two lines are parallel if they have the same slope.
The problems in this section (and the next) are cool because they pull together a bunch of stuff I've shown you.
Check it out:
Let's find the equation of the line that passes through the point
| ( -3 , 5 ) and is parallel to the line |  | . |
* To get the equation of a line we need:
 | a point -- which we have |
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| a slope -- which we can get from |  |
(since they are parallel, they'll have the same slope.)
STEP 1: Find the slope
STEP 2: Use the point-slope form with the point
( -3 , 5 ) and the slope of 2
I'll let you check our answer by graphing both lines to see if they are really parallel!
| Graph |  | and |  |
on the same graph.
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YOUR TURN:
Find the equation of the line that passes through the point ( 1 , -2 ) and is parallel to the line
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Parallel
From Greek: para allelois "beside one another"
Lines are parallel if they lie in the same plane, and are the same distance apart over their entire length
Try this Drag any orange dot at the points P or Q. As the line PQ moves, the line RS will remain parallel to it.
Parallel lines remain the same distance apart over their entire length. No matter how far you extend them, they will never meet.
The arrows
To show that lines are parallel, we draw small arrow marks on them. In the figure above, note the arrows on the lines PQ and RS. This shows that these lines are parallel. If the diagram has another set of parallel lines they would have two arrows each, and so on.
Shorthand notation
When we write about parallel lines there is a shorthand we can use. We can write  which is read as "the line segment PQ is parallel to the segment RS".
Recall that the horizontal bar over the letters indicates it is a line segment.
Constructing a parallel line
In the Constructions chapter, there is an animated demonstration of how to construct a line parallel to another that passes through a given point, using only a compass and straightedge. See Constructing a parallel line through a point.
Parallel planes
 In a very similar way, planes can be parallel to each other also. It means that the two planes are the same perpendicular distance apart everywhere. So, for example, the cards in a deck of cards are parallel.
An example of this is a cylinder, where the two bases (ends) are always parallel to each other. 
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