To solve this problem, we can use the concept of similar triangles. Since $AB \parallel CD$, triangles $ABE$ and $CDE$ are similar by the basic proportionality theorem (also known as Thales' theorem).

Given Information:

1. $AB \parallel CD$
2. $AB = 4$, $CD = 6$, and $BC = 15$

To Find:

The length of $EC$.

Solution:

Since triangles $ABE$ and $CDE$ are similar, their corresponding sides are proportional:

$\frac{AB}{CD} = \frac{BE}{DE} = \frac{AE}{EC}$

Substitute the known values:

$\frac{AB}{CD} = \frac{4}{6} = \frac{2}{3}$

Let $EC = x$ and $AE = y$. Then: