To solve the problem, we need to find the measure of $\angle BCG$ given that $AB \parallel EF$, and $CG$ bisects $\angle BCD$. We are also given that $\angle DEF = \frac{3\pi}{5}$.

Here’s the step-by-step solution:

1. Identify Parallel Lines and Angles: Since $AB \parallel EF$, corresponding angles formed with any transversal are equal. In particular, let’s identify the relationship involving $\angle DEF$ and angles on the transversal $CD$.

2. Angles on the Parallel Lines: Given $\angle DEF = \frac{3\pi}{5}$, and considering that $CG$ is the angle bisector of $\angle BCD$, we should look at how this angle interacts with $CD$.

3. Use the Angle Bisector Property: Since $CG$ bisects $\angle BCD$, we have: $\angle BCG = \frac{1}{2} \times \angle BCD$

4. Determining $\angle BCD$: To find $\angle BCD$, notice that since $\angle DEF = \frac{3\pi}{5}$ and $EF \parallel AB$, $\angle DEF$ and $\angle BCD$ are corresponding angles. Therefore, $\angle BCD = \frac{3\pi}{5}$

5. Calculating $\angle BCG$: With $\angle BCD$ determined, we can find $\angle BCG$: $\angle BCG = \frac{1}{2} \times \angle BCD = \frac{1}{2} \times \frac{3\pi}{5} = \frac{3\pi}{10}$

So, the measure of $\angle BCG$ is $\boxed{\frac{3\pi}{10}}$.

Would you like more details on this solution, or do you have any questions? Here are some questions you might want to ask:

1. How can you determine the corresponding angles when parallel lines are involved?
2. What is the significance of an angle bisector in a geometry problem?
3. How do parallel lines affect the angles formed by a transversal?
4. Can you provide another example where angle bisectors are used to find specific angles?
5. What are some key properties of parallel lines in geometry?

Tip: Always keep in mind the relationships and properties of parallel lines and transversal angles, as they are crucial in solving many geometry problems.