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To find the angle between \( GB \) and the plane \( ABCD \), we need to consider the geometric relationships in the given figures. Here's a step-by-step solution:

1. **Understand the Configuration:**
   - \( ABCD \) is a square with side length 12 cm.
   - \( EF \) is a fold line parallel to \( AD \) such that \( AE : EB = 1 : 2 \).
   - \( \angle AEB = 80^\circ \).
   - \( G \) is the midpoint of \( AD \), and \( J \) is the projection of \( G \) on the table.

2. **Key Points and Measurements:**
   - Since \( AD = 12 \) cm, \( AE = \frac{12}{3} = 4 \) cm and \( EB = \frac{12 \times 2}{3} = 8 \) cm.
   - Since \( G \) is the midpoint of \( AD \), \( AG = GD = 6 \) cm.
   - \( \angle AEB = 80^\circ \) and \( AB = 12 \) cm.

3. **Projection on the Table:**
   - \( J \) is directly below \( G \) on the table, vertically aligned.
   - \( G \) is above the midpoint \( M \) of \( EF \).

4. **Calculate the Height of \( G \) Above the Table:**
   - \( \Delta AEB \) is folded along \( EF \), forming a 3D configuration. The height \( h \) from \( G \) to the table can be calculated using the right triangle \( \Delta GJ \).

5. **Determine the Angle Between \( GB \) and the Plane \( ABCD \):**
   - We use the fact that \( G \) is at the midpoint of \( AD \), and \( \angle AEB = 80^\circ \) helps us understand the folding.
   - Given the symmetry and folding, \( GB \) forms a vertical line. 

To find the angle \( \theta \) between \( GB \) and the plane \( ABCD \):

- When the paper is folded, \( G \) is at height \( h \) above the plane \( ABCD \).
- The plane \( ABCD \) is horizontal, and \( GB \) is a line connecting \( G \) to \( B \).

Using the fact that \( \Delta AEB \) makes an angle \( 80^\circ \) with \( E \):

1. \( GB \) is perpendicular to the plane \( ABCD \) since it forms a right triangle in the vertical direction from the midpoint \( G \) to \( B \).

Therefore, the angle \( \theta \) between \( GB \) and the plane \( ABCD \) is \( 90^\circ \).

**Conclusion:**
\[
\boxed{90^\circ}
\]

This is because \( GB \) is a vertical line from point \( G \) to point \( B \), and the plane \( ABCD \) is horizontal, making the angle between them \( 90^\circ \).

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Would you like further details or have any questions about the solution?

Here are five questions you can ask:
1. How do you find the distance between point G and the table?
2. What is the length of segment \( EF \)?
3. How do you find \( \angle EAB \) in Figure 2?
4. What are the coordinates of point J in the projection?
5. How does the folding line \( EF \) affect the angles in the figure?

**Tip:** When dealing with problems involving planes and lines in 3D geometry, visualize the configuration and use right-angle triangles to simplify the calculation of angles and distances.

Understand how to find the angle between line segment GB and the plane ABCD in a 3D geometry problem. Detailed step-by-step solution with geometric configurations and angle calculations.

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