Question

If two segments are parallel, then they must be coplanar.

Answer 1

Answer:

By definition, two lines are coplanar iff they lie on the same plane. We want to show that if two lines are parallel →→ the two lines are also coplanar. Suppose the lines are non-trivially parallel (they are not the same line).

Take two parallel lines ΔΔ and Δ′Δ′. Take two points AA and B∈ΔB∈Δ and another one CC ∈Δ′∈Δ′. This forms a plane P=(A,B,C)P=(A,B,C). If we can show that every point in ΔΔ is in PP and every point in Δ′Δ′ is in PP then we effectively show that both ΔΔ and Δ′Δ′ are in P, making them coplanar, proving our claim.

Take any point X∈ΔX∈Δ. Trivially, because XX is in Δ=(AB)Δ=(AB), then X∈P=(A,B,C)X∈P=(A,B,C).

Now take any point Y∈Δ′Y∈Δ′. If it is ∈P∈P, then we are done. If it is not in PP, then it must be true that the line Δ′Δ′ intersects PP at a single point, namely CC. In a sense, Δ′Δ′ pierces the plane PP containing