Proving two versions of Pasch's Axiom are equivalent

Question

I am trying to show that these two versions of Pasch's axiom are the same.

A1. If a line enters a triangle at a vertex, then the line intersects the opposite side.

A2. If a line enters a triangle at a side without intersecting the opposite vertex, then the line intersects one of the other two sides.

But, I can figure out how to use either axiom for proving the other axiom.

Answer 1

I'll just do one direction. Assume A1, we work to show A2. Let $ABC$ be a triangle, and assume we have a line $l$ entering triangle $ABC$ at side $AB$ (without loss of generality) at a point $D$ on $AB$ and not intersecting $C$. Then $l$ enters either triangle $DAC$ or $DBC$ at vertex $D$, and then apply A1 to whichever the case is (e.g. if $l$ enters $DAC$ then A1 says it intersects $AC$.