Unizor - Geometry3D - Lines and Planes - Parallel Lines

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Two straight lines in three-dimensional space are called parallel if
(1) they lie in the same plane and
(2) they do not intersect.

Exercise 1:
Construct a straight line l in three-dimensional space parallel to a given straight line a and passing through a given point P outside of a given line.
Solution:
According to one of the axioms that we discussed in previous lectures, we can construct a plane γ passing through a given line a and a given point P. It exists and is unique, according to this axiom. Then we construct a straight line l parallel to line a and passing through point P within this plane γ using the techniques known from two-dimensional geometry. Obtained line l is a line parallel to a given line a because (1) it lies in the same plane γ with line a and (2) does not intersect a (since it's parallel within plane γ).

Exercise 2:
Prove that there is one and only one line parallel to a given line a and passing through a given point P outside of a given line.
Proof:
The existence of one such line l follows from the construction exercise above.
To prove the uniqueness of such a constructed line, assume there is another line m also parallel to a, also passing through point P.
Since, by definition of parallel lines, lines a and m lie in the same plane (let's call it δ), we have a situation that there are two planes, γ (as constructed to obtain line l) and δ (containing lines a and m), both of which contain a given line a and a given point P. According to one of the axiom, there is one and only one plane that passes through a given line and a given point outside it. So, plains γ and δ are identical, that is they constitute one and the same plane. But in this plane, according to axioms of two-dimensional geometry, there is one and only one line parallel to a given line and passing through a point outside it. Therefore, lines l and m are identical.
That proves the uniqueness of a line parallel to a given line and passing through a given point outside it in the three-dimensional space.

Important part of a definition of parallel lines is the requirement that they must lie in the same plane. Without this requirement two lines might lie in space without intersecting each other and not be parallel in any sense. For instance, a road and a bridge over it might represent such two lines. They do not intersect but cannot possibly be considered as parallel. These non-parallel non-intersecting lines in three-dimensional space are called skew lines.