To solve the problem, we need to analyze the given geometric configuration and apply relevant geometric properties and theorems. Let's proceed step-by-step:

1. Given Information:

   • $AB \parallel EF$
   • $CG$ bisects $\angle BCD$
   • $\angle DEF = \frac{3\pi}{5}$

2. Objective:

   • Find $\angle BCG$ and express the result in terms of $\pi$.

Step-by-Step Solution:

1. Parallel Lines and Transversal: Since $AB \parallel EF$, and $CD$ is a transversal, we have: $\angle ACD = \angle EFD$ This is due to corresponding angles being equal when a transversal intersects two parallel lines.

2. Bisector Property: Since $CG$ bisects $\angle BCD$, we have: $\angle BCG = \angle GCD$ Let $\angle BCG = \theta$. Therefore, $\angle GCD = \theta$.

3. Angles Sum Around Point: The angles around point $D$ add up to $2\pi$. We can write: $\angle BCD = \angle BCG + \angle GCD = 2\theta$

4. Angle DEF Calculation: Given $\angle DEF = \frac{3\pi}{5}$, we need to find the remaining angle around point $E$. Since $E$ is a point on $CD$ and $AB \parallel EF$, we need to consider the supplementary angles around $E$: $\angle DEC + \angle DEF = \pi$ $\angle DEC = \pi - \frac{3\pi}{5} = \frac{2\pi}{5}$

5. Sum of Angles at Point D: Considering the angles around point $D$: $\angle BCD + \angle DEF = 2\theta + \frac{3\pi}{5}$ But we also need to consider $\angle EFD$: $\angle EFD = \angle ACD = \frac{2\pi}{5} \, \text{(from step 4)}$

6. Angle Sum Around D: The total angle sum around point $D$ should be: $2\theta + \frac{2\pi}{5} + \frac{3\pi}{5} = 2\pi$ $2\theta + \pi = 2\pi$ $2\theta = \pi$ $\theta = \frac{\pi}{2}$

Final Answer:

Thus, $\angle BCG = \frac{\pi}{2}$.

Would you like more details or have any questions?

Here are some questions you might ask:

1. How do you determine the angles around a point?
2. Can you explain the property of angle bisectors in more detail?
3. Why do the angles around a point sum up to $2\pi$?
4. How does the parallel line property help in solving geometric problems?
5. What are the key theorems used in solving this problem?

Tip: Always check the relationship between angles and lines, especially when dealing with parallel lines and transversals, as they often provide helpful angle properties.