https://file.aiquickdraw.com/mathbot/file/1721958925732i4erpeul.png

$$ {{\displaylines15\text{. (}5分)\,如图,\,点C,\,点E分别为AB,\,CD上一点,\,已知AB//EF,\,且CG平分\angle BCD.\,若\angle DEF=\frac{3\pi}{5},\,则\angle BCG\\ =_{--------.}\left(结果保留\pi\right)}} $$

Given the problem and the provided diagram, let's solve for $\angle BCG$.

1. **Given Information:**
    - $ AB \parallel EF $
    - $ CG $ bisects $\angle BCD$
    - $\angle DEF = \frac{3\pi}{5}$

2. **Step-by-Step Solution:**
    - Since $ AB \parallel EF $, the alternate interior angles are congruent. Thus, $\angle BAC = \angle AEF = \angle DEF = \frac{3\pi}{5}$.

    - Because $ CG $ is the angle bisector of $\angle BCD$, it divides $\angle BCD$ into two equal parts:
      \[
      \angle BCG = \frac{1}{2} \angle BCD
      \]

3. **Determine $\angle BCD$:**
    - Knowing $ \angle BCG $ is half of $\angle BCD$, and $\angle BCG = \angle AEF$ since $ CG $ bisects $\angle BCD$,
      \[
      \angle BCD = 2 \angle AEF
      \]
    - Substitute $\angle AEF = \frac{3\pi}{5}$,
      \[
      \angle BCD = 2 \times \frac{3\pi}{5} = \frac{6\pi}{5}
      \]

4. **Calculate $\angle BCG$:**
    - Since $\angle BCG$ is half of $\angle BCD$,
      \[
      \angle BCG = \frac{1}{2} \times \frac{6\pi}{5} = \frac{3\pi}{5}
      \]

Therefore, $\angle BCG = \frac{3\pi}{5}$.

Would you like any more details or have any further questions?

Here are some additional questions you might find interesting:
1. How would the solution change if $AB$ were not parallel to $EF$?
2. Can you explain how alternate interior angles work when two lines are parallel?
3. What would happen if $\angle DEF$ was not given? How would you approach finding $\angle BCG$?
4. How does the concept of angle bisectors help in solving geometric problems?
5. Can you find $\angle DCF$ given the above information?

**Tip:** When dealing with parallel lines, always look for alternate interior angles and corresponding angles to find unknown angle measures.

Explore how to solve a geometric problem involving the calculation of angle BCG. Given AB parallel to EF and CG bisects angle BCD, find angle BCG with a detailed step-by-step solution. Suitable for high school students studying geometry.

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