# Parallel Lines and Ratio Proof

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

$\displaystyle \frac{AB}{BC} = \frac{DE}{EF}$.

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

$\frac{AB}{BC} = \frac{AG}{GH}$ (1)

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

$AG = DE$ and $GH = EF$. (2)

From (1) and (2),

$\displaystyle \frac{AB}{BC} = \frac{DE}{EF}$

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.