It can be introduced. Line-plane parallelism can only say that the intersection line between a plane and this plane is parallel to this line, and all lines in this plane that are parallel to this intersection line are also parallel to this line. In face-to-face parallelism, it is also possible that the lines in these two planes are opposite.

Theorem 1

If a line is parallel to a plane, then the intersection of any plane through which the line passes with the plane is parallel to the line.

It is known that: a∥α, a∈β, α∩β=b. Prove: a∥b

Proof: assuming that a and b are not parallel, let their intersection be P, i.e., P is on the line a, b.

∵ b ∈ α, ∴ a ∩ α = P

Contradiction with a∥α

∴ a∥b

This theorem reveals that parallelism of a line to a plane implies parallelism of a line to a line. By making a line parallel to the plane one gets a line parallel to the line. This gives an important method of making parallel lines.

Note: Just because a line is parallel to the plane does not mean that it is parallel to all lines in this plane, but if a line is perpendicular to the plane, then this line is perpendicular to all lines in this plane.

Theorem 2

A line is parallel to a plane, then the line is perpendicular to the perpendicular of this plane.

It is known that: a∥α, b⊥α. Prove that: a⊥b

Proof: Since there are an infinite number of perpendicular lines to α, b can be translated to intersect a. Let the translated line be c, a∩c=M, and the foot of the perpendicular line between c and α is N.

∵ Two intersecting lines define a plane

∴ Let the plane formed by a and c be β, and α∩β=l

∵ N ∈ c, N ∈ α, c?β

∴ N ∈ l, and by Theorem 1 a ∥ l

∵ c ⊥ α, l?α

∴ c ⊥ l

∴ a ⊥ c

Since translations don't change the direction of the line, it follows that a ⊥ b