Geometry for Elementary School/Parallel lines
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| Geometry for Elementary School | ||
| Some impossible constructions | Parallel lines | Squares |
[edit] Definition
The definition of parallel lines is based on Book I, definition 23.
Parallel lines are straight lines that never intersect, which means that they never cross. Notice that when we look at parallel parts of shapes there is no place where they intersect even if we extend the lines.
[edit] The parallel lines postulate
The postulate appears in Euclid’s elements as the fifth postulate.
Let there be two lines. If there is a third line that intersects them such that the sum of the interior angles on one side is smaller than two right angles then the two lines intersect.
This postulate was suspect as redundant. Mathematicians though that instead of assuming it, the postulate can be deduced from other postulates. However, the attempts to deduce this postulate failed. The reason to this failure is that the indeed, the parallel postulate doesn’t follow from the other ones. While we assume it in plane geometry, one can define different geometries (e.g., on a ball) for which this postulate is not valid.
