Parallel lines have many important properties that you have to memorize in order to be successful in proving problems. In this post, we will discuss the properties of parallel lines intersecting two other lines. We are going to show the relationship among three parallel lines intersecting two lines.

In the figure below, parallel lines *l*, *m*, and *n*, intersect line *a* at points *A*, *B*, and *C*, and intersect line *b* at points *D*, *E*, and* F*. The theorem is that the ratio of *AB* to *BC* is the same as the ratio of *DE* to *EF*.

**Theorem** If parallel lines* l*, *m*, *n*, intersect line *a* at *A*, *B*, *C* respectively and intersect line *b* at *D*, *E*, and *F*, respectively, then

.

**Proof**

First, we draw a line parallel to *b* and passing through *A* as shown below. Let’s name the intersections of that line and* m* and n as *G* and *H* respectively.

Now in triangle *ACH*, since *BG* is parallel to *CH*, we know that

(1)

Since quadrilaterals *AGED* and *GHFE* are parallelograms, their opposite sides are congruent. Therefore,

and . (2)

From (1) and (2),

which is what we want to prove.

As an exercise related to the theorem above, you may want to prove the following special cases.

(1) a and b are parallel

(2) A and D are the same point.