Parallel lines in the plane and in space

The plane direct are called parallel if they have no common points, i.e. they do not intersect. To indicate the parallel use of the special badge || (parallel lines a || b).

For straight, lying in the space, the lack of common points is not enough – so they in space are parallel, they must belong to the same plane (otherwise they will be skewed).

For examples of parallel lines do not go away, they accompany us everywhere, in the room – this is the line of intersection of walls with ceilings and floors, on the notebook sheet – the opposite edge, etc.

It is obvious that having a parallelism of two straight lines and a third line parallel to one of the first two, it will be parallel and the second.

Parallel lines on plane bound by the statement which is not proved using the axioms of plane geometry. It is taken as fact, as an axiom: for any point on the plane, not lying on a straight line, there is only one straight line which passes through it parallel to this. This axiom known to every sixth-grader.

Its spatial generalization, that is the statement that for any point in space, not lying on a straight line, there is only one straight line which passes through the parallel to this, it can easily be proved by using the already known axioms of parallelism on the plane.

Properties of parallel lines

• If any of the parallel two lines are parallel to a third, then they are mutually parallel.

This property is possessed parallel lines and planes, and space.
as an example, let us consider its rationale in the solid geometry.

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For example the parallel direct b and direct a.

The Case when all lines lie in the same plane leave the plane.

Suppose a and b belong to the plane of the betta, and gamma-plane, which belong to a and C (by definition of parallelism in direct space should belong to the same plane ).

Assuming that the plane of the beta and gamma of different and mark on line b from the plane of betta some point B, a plane drawn through the point B and the line must cross the plane of the betta in a straight line (let's call it b1).

If the received line b1 crosses the plane of the gamma, on the one hand, the point of intersection would have to lie in a, because b1 belongs to the plane of the betta, and on the other , it should belong to and, since b1 belongs to the third plane .
But parallel lines a and C should not overlap.

Therefore, line b1 must belong to the plane of the betta and it does not have common points with a, therefore, according to the axiom of parallelism, it coincides with b.
We received is coincident with the straight line b, b1, which belongs to the same plane with the straight line and thus it does not intersect, i.e. b and-parallel

• Through the point that does not lie on a given straight line, parallel to this can take place only one single line.

• Lying in the plane perpendicular to the third two lines are parallel.

• When the condition crosses one of the parallel two lines the same plane crosses the second line.

• Matching and pattern lying inside corners formed by intersection of two straight lines parallel to a third are equal, the amount of formed when the inner unilateral equal to 180°.

True and return statements that can be mistaken for signs of parallelism of two straight lines.

The Condition of parallel lines

The above properties and characteristics are the conditions of parallel lines, and they can be proved using geometry. In other words, to prove the parallelism of the two existing lines is sufficient to prove their third straight parallelism or equality of angles, whether it's relevant or crosswise lying, etc.

For the proof is mainly used the method of “nasty”, that is with the assumption that the direct non-parallel. Based on this assumption, it is easy to show that in this case violated the specified conditions, for example, cross lying the interior angles are equal, which proves the incorrectness of the assumptions made.