Congruent Parts of Parallel Lines

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

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. Since $\overline{UW}$ is congruent to $\overline{WY},$ their lengths are equal, making the quotient $\frac{UW}{WY}$ equal to $1.$ Because of this, the length of $\overline{VX}$ is equal to the length of $\overline{XZ}.$ Since their lengths are the same, $\overline{VX}$ is congruent to $\overline{XZ}.$ 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.