The Proportional Segments Theorem

By repeated applications of the Triangle Midsegment Theorem, we can arrive at more general results:

**The Proportional Segments Theorem:**

*Three or more parallel lines cut any two transversals into proportional segments.*

For example, in the following figure, :

**Sample Problem:** Find *x*:

**Solution:** Set up the proportion: . Then cross-multiply and solve:

, so

**Corollary:**

*If a segment with endpoints on two sides of a triangle is parallel to the third side, it divides the two sides into proportional segments.*

**Sample Problem:** Find *x*:

**Solution:** This is the same as the last problem, as can by seen by drawing a third parallel line at the top vertex of the triangle:

Therefore, it has the same solution: , so *x* = 9.