1. The problem statement, all variables and given/known data Given P not on line l, line PQ perpendicular to l at Q, line m perpendicular to line PQ at P, and point A ≠ P on m. Let ray PB be the last ray between ray PA and ray PQ that intersects l, B being the point of intersection. There exists a point C on l such that Q*B*C. It follows that ray PB is not the last ray between ray PA and ray PQ that intersects l, and hence all rays between ray PA and ray PQ meet l. Thus, m is the only parallel to l through P. 2. Relevant equations Betweenness axioms Definition of plane separation 3. The attempt at a solution I'm pretty sure there's something not quite right with the statement that because Q*B*C, that QC is between rays PQ and PA. I feel like we're assuming something and that the flaw is in this statement.