Unizor - Geometry3D - Lines and Planes - Line Parallel Plane

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Theorem 1. If a straight line a lying outside of a plane γ is parallel to a straight line b that lies in a plane γ then line a and plane γ are parallel.

Theorem 2. If a straight line a lying outside of a plane γ is parallel to this plane and another plane δ contains line a and intersects plane γ, the straight line b of intersection of planes γ and δ is parallel to line a.

Theorem 3. Given a straight line a in three-dimensional space and a point P outside it.
Prove that there is one and only one line b that passes through point P and is parallel to line a.
Notice: Here we deal with three-dimensional space, not with the plane where this statement is, actually, a form of the 5th Euclid's postulate.

Theorem 4. Given a straight line a that is parallel to two intersecting planes γ and δ.
Prove that line a is parallel to line b of intersection between γ and δ.

Theorem 5. Given a straight line a in 3D space and two other lines, m and n, each parallel to line a.
Prove that lines m and n are parallel to each other.