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

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

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.

Explore how to solve a geometry problem involving parallel lines, transversals, and angle bisectors. Detailed step-by-step solution provided with the use of corresponding angles and the angle bisector theorem. Suitable for high school students.

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