On the plane, straight lines are called parallel if they do not have common points, that is, they do not intersect. To indicate parallelism use the special icon || (parallel lines a || b).
For lines lying in space, the requirement of the absence of common points is not enough - for them to be parallel in space, they must belong to the same plane (otherwise they will be crossed).
We don’t have to go far for examples of parallel lines, they accompany us everywhere, in the room there are lines of intersection of the wall with the ceiling and the floor, on the notebook sheet are opposite edges, etc.
It is quite obvious that, having the parallelism of two lines and the third straight line parallel to one of the first two, it will be parallel to the second.
Parallel straight lines on the plane are connected by a statement that cannot be proved using the axioms of planimetry. It is taken as a fact, as an axiom: for any point on a plane that does not lie on a line, there is a single line that passes through it parallel to the given one. Every sixth grader knows this axiom.
Its spatial generalization, that is, the assertion that for any point in space that does not lie on a straight line, there is a single straight line that passes through it parallel to the given one, can be easily proved using the axiom of parallelism on the plane already known to us.
Properties of parallel lines
- If any of the parallel two lines is parallel to the third, then they are mutually parallel.
This property is possessed by parallel lines both on the plane and in space.
As an example, consider its justification in stereometry.
Suppose that lines b and c are a.
The case when all the lines lie in the same plane will leave planimetry.
Suppose a and b belong to the betta plane, and gamma is the plane to which a and c belong (by definition of parallelism in space, the lines should belong to one plane).
If we assume that the betta and gamma planes are different and mark a point B on line b from the betta plane, then the plane drawn through point B and line c should intersect the betta plane in a straight line (we denote it by b1).
If the resulting line b1 intersected the gamma plane, then, on the one hand, the intersection point would have to lie on a, since b1 belongs to the betta plane, and on the other, it should belong to c, since b1 belongs to the third plane.
But parallel lines a and c should not intersect.
Thus, the line b1 must belong to the betta plane and at the same time not have common points with a, therefore, according to the parallelism axiom, it coincides with b.
We have received the straight line b1, which coincides with line b, which belongs to the same plane with 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 parallel to a given line, only one single line can pass.
- Two straight lines lying on a plane perpendicular to the third are parallel.
- Provided that the plane of one of the parallel two lines intersects, the second line intersects the same plane.
- Corresponding and crosswise lying internal angles formed by the intersection of parallel two straight lines of the third are equal, the sum of the resulting internal one-sided is 180 °.
The converse statements, which can be taken as signs of parallelism of two lines, are also true.
The condition of parallel lines
The properties and features formulated above are the conditions for parallel lines, and they can be quite proved by the methods of geometry. In other words, to prove the parallelism of two existing lines, it is enough to prove their parallelism to the third line or the equality of angles, whether corresponding or lying crosswise, etc.
To prove this, they mainly use the “on the contrary” method, that is, on the assumption that the lines are not parallel. Based on this assumption, it can easily be shown that in this case the specified conditions are violated, for example, the lying internal angles are unequal, which proves the incorrectness of the assumption made.