IV - Parallels Without a Common Perpendicular

Abstract

This chapter discusses the parallels without a common perpendicular. It presents a theorem that states that given any line g and any point F not on it, there exist exactly two lines m, n which go through F, are parallel to g, and do not have a common perpendicular with g. If E is the projection of F on g, then m, make equal acute angles α, β with line EF. Within one pair of vertical angles formed by m and n lie all the lines through F which meet g; within the other pair of vertical angles lie all the parallels to g through F which have a common perpendicular with g. The chapter presents various theorems concerning parallel lines without a common perpendicular and their proofs. The chapter discusses properties of boundary parallels. A line that subdivides an angle of a trilateral meets the opposite side. It presents a theorem that states that a line which meets a side of a trilateral but does not go through a vertex will meet another side, provided that the line is not a boundary parallel to an outer side.