Parallel Lines and Proportional Parts

When two lines cross a set of parallel lines, the parallels slice both lines into matching pieces. Those pieces are not usually equal — but the ratio between them is always the same on each line.

Theorem Proportional Parts

If three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

The two transversals are cut into corresponding pieces, so:

\frac{AB}{BC} = \frac{A'B'}{B'C'}

Note Two things to notice

The ratio does not depend on the angle at which the transversals cross the parallels. Slide the lines to any tilt and $\tfrac{AB}{BC}$ stays fixed, so $\tfrac{A'B'}{B'C'}$ follows.

The equal-spacing case is the most useful one: if the parallels cut off equal pieces on one transversal, then $AB = BC$ forces $A'B' = B'C'$ — the ratio is simply 1. This is exactly why the single ticks match on both sides of the figure above.

Example Using the proportional parts theorem

Three parallel lines cut two transversals. On the first transversal the two pieces are equal. On the second transversal they measure $4x+3$ and $6x-5$ . Find $x$ .

Equal pieces on one line ⟹ ratio = 1 ⟹ the matching pieces are equal.

4x + 3 = 6x - 5

8 = 2x

x = 4

Check:

4(4)+3 = 19

and

6(4)-5 = 19

. ✓

⟹ x = 4, and each piece is 19 units.

Summary

Three or more parallel lines cut two transversals into proportional pieces: $\tfrac{AB}{BC} = \tfrac{A'B'}{B'C'}$ .
The ratio is independent of the angle at which the transversals cross.
If one transversal is cut into equal pieces, so is the other — the ratio is 1.
This theorem is the foundation of the triangle proportionality theorem.