Congruent Parts of Parallel Lines

Let r, s, and t be three parallel lines that cut transversal lines l and m. Suppose that the parallel lines cut two congruent segments on transversal line l.

By the Three Parallel Lines Theorem, it is known that the parallel lines divide the transversal proportionally. This proportion can be written as an equation. UW/WY = VX/XZ Since UW is congruent to WY, their lengths are equal, making the quotient UWWY equal to 1. Because of this, the length of VX is equal to the length of XZ. 1 = VX/XZ ⇒ XZ = VX Since their lengths are the same, VX is congruent to XZ. VX ≅ XZ This relation can be shown in the graph.

It should be noted that this theorem can be proven similarly for a different transversal or for more parallel lines.