Parallel lines are important concepts in axiomatic geometry. The parallel axiom of Euclidean geometry can be equivalently expressed as "there exists a unique true straight line and the known true straight line intersects a point outside the straight line". However, its negative form "there is no true straight line parallel to the known true straight line at a point outside the true straight line" or "there are at least two true straight lines and known true straight lines at a point intersecting with the true straight line" can be regarded as the deaf generation of Euclidean geometry axiom, and non-Euclidean geometry independent of Euclidean geometry has been developed.

If both lines are parallel to the third line, then the two lines are also parallel to each other. If a ||| b, b||c, then a |||| c.

Definition:

Two straight lines that never intersect in the same plane are called parallel lines. Parallel lines must be defined on the same plane, which does not apply to solid geometry, such as straight lines on different planes, which are neither intersecting nor parallel.

In advanced mathematics, the definition of parallel lines is that two lines intersecting at infinity are parallel lines, because there is no absolute parallelism in theory.

For the judgment of parallel lines, it is a conclusion that two lines are parallel, but for the nature of parallel lines, it is a condition that two lines are parallel. It is known that two straight lines are parallel. The relationship between angles obtained from parallel lines is the nature of parallel lines, including: ① two straight lines are parallel and the same angle is equal; ② Two straight lines are parallel and the internal dislocation angles are equal; ③ The two straight lines are parallel and complementary.