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Rule

Congruent Parts of Parallel Lines

If three or more parallel lines cut congruent segments on a transversal, then the parallel lines cut congruent segments on any transversal.

Three parallel lines and two transversals - congruent segments

Applying the theorem to the diagram above, the following can be written.

If $r ∥ s ∥ t$ and $\overline{U W} ≅ \overline{W Y},$ then $\overline{V X} ≅ \overline{X Z} .$

Proof

Proof of the Congruent Parts of Parallel Lines Theorem

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.

\frac{U W}{W Y} = \frac{V X}{X Z}

Since

\overline{U W}

is congruent to

\overline{W Y},

their lengths are equal, making the quotient

\frac{U W}{W Y}

equal to

1 .

Because of this, the length of

\overline{V X}

is equal to the length of

\overline{X Z} .

1 = \frac{V X}{X Z} \Rightarrow X Z = V X

Since their lengths are the same,

\overline{V X}

is congruent to

\overline{X Z} .

\overline{V X} ≅ \overline{X Z}

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.

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Proof