the plane lines are called parallel if they do not have common points, that is, they do not intersect.To indicate parallelism using a special icon || (parallel lines a || b).
to lines lying in the space requirements of a lack of common points is not enough - so they are parallel in space, they must belong to the same plane (otherwise they would skew).
For examples of parallel lines do not have to go far, they accompany us everywhere in the room - a line of intersection of the walls to the ceiling and the floor, on the notebook sheet - the opposite edges, etc.
It is evident that having two parallel lines and a third line parallel to one of the first two, it will be parallel to the second.
parallel lines on the plane bound statement is not proved using the axioms of plane geometry.It is taken as a fact, as an axiom: for any point on the plane not lying on a straight line, there is a unique line that passes through it parallel to this.This axiom knows every sixth grader.
its spatial generalization, that is, the claim that for every point in space, not lying on a straight line, there is a unique line that passes through it parallel to this, is easily proved by the already known to us on the plane parallel axiom.
properties of parallel lines
- If any of the two parallel lines parallel to a third, then they are parallel.
have this property, and parallel lines on the plane and in space.
For example, consider its rationale in the solid geometry.
Let parallel lines b and c direct a.
case where all the lines lie in the same plane leave the plane geometry.
Assume, a and b belong to plane beta and gamma - plane, which holds a and c (for the definition of parallel lines in space should belong to the same plane).
Assuming that the plane beta and gamma and different note on the line b in the plane of the beta certain point B, the plane through the point B, and to direct the plane to cross the betta in a straight line (denoted by b1).
If obtained b1 line intersects the plane of gamma, is on the one hand, the intersection point should lie on a as b1 belongs beta plane, and on the other, it should belong and, since b1 belongs to a third plane.
But parallel lines a and should not overlap.
Thus, the lines b1 should belong to the plane of the beta and do not have common points with a, it follows, according to the axiom of parallelism, it coincides with the b.
We received coincides with the line b line b1, which is owned by the same plane with the straight line with and at the same time it does not intersect, that is, b and c - parallel
- A point that is not on a given line parallel to this mayIt takes only one unique line.
- lying on a third plane perpendicular to two straight parallel.
- Provided intersection of the plane of one of the two parallel lines, the same plane and crosses the second line.
- appropriate and cross lying inside corners formed by the intersection of two straight lines parallel to a third are equal to the sum formed from the one-sided to the internal is 180 °.
converse is also true, which can be mistaken for signs of parallelism of two lines.
Parallelism condition of straight
stated above properties and attributes are the conditions of parallel lines, and it is possible to prove the methods of geometry.In other words, to prove the parallelism of the two existing lines is sufficient to prove their third straight parallel or equality of angles, whether relevant or cross lying, etc.
To prove the method is mainly used "on the contrary", that is with the assumption that the lines are not parallel.Based on this assumption, it is easy to show that in this case violated the specified conditions, such as cross lying inside corners are not equal, which proves incorrect assumptions made.
