Three Parallel Lines Theorem

In the diagram, draw $\overline{V Y}$ and let $P$ be the point of intersection between this segment and line $s .$

Next, separate

△ U V Y

and

△ Y Z V .

Since

\overline{W P} ∥ \overline{U V},

by the Triangle Proportionality Theorem

\overline{W P}

divides

\overline{Y U}

and

\overline{Y V}

proportionally.

\frac{U W}{W Y} = \frac{V P}{P Y}

Applying the same reasoning in

△ Y Z V,

it can be said that

\overline{P X}

divides

\overline{V Y}

and

\overline{V Z}

proportionally.

\frac{V P}{P Y} = \frac{V X}{X Z}

Finally, the Transitive Property of Equality leads to the desired proportion, showing that the three parallel lines divide the transversals proportionally.

⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎧ \frac{U W}{W Y} = \frac{V P}{P Y} \frac{V P}{P Y} = \frac{V X}{X Z} \Rightarrow \frac{U W}{W Y} = \frac{V X}{X Z}

Three Parallel Lines Theorem

Proof

Recommended exercises