Parallel lines in the plane and in space

On a plane, lines are called parallel if they do not have common points, that is, they do not intersect. To denote parallelism, use the special icon || (parallel lines a || b).

For lines lying in space, the requirementsthe absence of common points is not enough - so that they are parallel in space, they must belong to the same plane (otherwise they will be interbreeding).

It is not necessary to go far beyond examples of parallel straight lines, they accompany us everywhere, in the room - these are the lines of intersection of the wall with the ceiling and the floor, on the tetrad sheet - opposite edges, etc.

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

Parallel lines on the plane are connectedan assertion that can not be proved with the help of planimetry axioms. It is taken as a fact, as an axiom: for any point on a plane not lying on a line, there is a single straight line that passes through it parallel to the given one. Every sixth grader knows this axiom.

Its spatial generalization, that is,the statement that for any point in space that does not lie on a straight line there is a unique straight line that passes through it parallel to a given one can be easily proved with the help of the axiom of parallelism already known to us on the plane.

Properties of parallel lines

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

This property is possessed by parallel lines both in the plane and in space.
As an example, let us consider its justification in stereometry.

Let us assume that b is parallel to a.

The case when all the lines lie in the same plane leave the planimetry.

Suppose a and b belong to the betta plane, and the gamma plane to which a and c belong (according to the definition of parallelism in space, the lines must belong to the same plane).

Assuming that the betta and gamma planesdifferent and mark a point B on the straight line b from the Betta plane, then the plane drawn through the point B and the line c must intersect the plane of the beta along the line (denoted by b1).

If the resulting straight line b1 intersected the planegamma, then, on the one hand, the point of intersection would have to lie on a, since b1 belongs to the betta plane, and on the other hand, it must belong to c, since b1 belongs to the third plane.
But in fact parallel lines a and c should not intersect.

Thus, the line b1 must belong to the betta plane and, in this case, do not have common points with a, hence, according to the axiom of parallelism, it coincides with b.
We have a line b1 coinciding with the straight line b, which belongs to the same plane with the straight line c and does not intersect it, that is, b and c are parallel

  • Through a point that does not lie on a given line, only one single line can pass parallel to a given line.
  • Lying on the plane perpendicular to the third two straight lines are parallel.
  • Given the intersection of the plane of one of the parallel two straight lines, the same plane intersects the second straight line.
  • Corresponding and cross-lying internal angles, formed by the intersection of parallel two straight third, are equal, the sum of the resulting internal one-sided ones is 180 °.

The converse statements, which can be taken as signs of the parallelism of two lines, are also true.

Condition of parallelism of lines

The properties and characteristics formulated aboverepresent the conditions for parallel lines, and they can be completely proved by methods of geometry. In other words, to prove the parallelism of two existing lines it is sufficient to prove their parallelism of the third straight line or the equality of angles, whether corresponding or crosswise, etc.

For the proof,"By contradiction," that is, with the assumption that the lines are not parallel. Proceeding from this assumption, it is easy to show that in this case the given conditions are violated, for example, cross-lying internal angles turn out to be unequal, which proves the incorrectness of the assumption made.

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