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Solve this using the method to solve geometry of 2D shapes

To solve the problem, we need to analyze the geometric properties of the 2D shape shown in the image. Let me break down the information given in the diagram:

1. **Triangle Relationships**: The shape consists of multiple triangles, with the angles, side lengths, and some key points marked.

2. **Given Angles**: 
   - The angle at point D is marked as \( 40^\circ \).
   - The angle at point B is marked as \( 36^\circ \).
   - The angle at point E is marked as \( 36^\circ \).
   - The angles at \( \angle AOD \) and \( \angle COF \) are labeled as \( x \).

3. **Given Side Lengths**:
   - Side \( AB = 6 \)
   - Side \( BC = 6 \)
   - Side \( CD = 6 \)
   - Side \( EF = 10 \)

4. **Triangles to Focus On**:
   - The green highlighted triangle \( AOD \).
   - The pink highlighted triangle \( COF \).

### Approach:
We will use the Law of Sines and Law of Cosines to find the unknowns in the diagram and use the relationships between the triangles to solve for the angle \( x \).

#### Steps:
1. **Use the Law of Sines**:
   - For triangle \( AOD \) with angle \( \angle AOD = x \) and known angles \( 40^\circ \) and \( 36^\circ \).
   - Similarly, for triangle \( COF \).

2. **Use the Law of Cosines**:
   - If necessary, to relate the sides and angles of the triangles.

3. **Apply Angle Properties**:
   - Since the sum of angles in a triangle is \( 180^\circ \), use this to find unknown angles.

4. **Determine the Value of \( x \)**:
   - Using the relationships derived from the above steps.

Let me calculate the necessary values now.The value of \( x \) for both angles \( \angle AOD \) and \( \angle COF \) is \( 104^\circ \).

### Explanation:
- The sum of the angles in any triangle is \( 180^\circ \).
- Given the angles \( 40^\circ \) and \( 36^\circ \) in both triangles \( AOD \) and \( COF \), the remaining angle \( x \) in both triangles is calculated as \( 104^\circ \).

Would you like further details or explanations on any specific part of this solution?

Here are five related questions you could explore next:
1. How would you use the Law of Sines to calculate unknown sides in the triangles?
2. What would happen to the angles if the side lengths of the triangles were altered?
3. How does the Law of Cosines help in solving such geometry problems?
4. How would this problem change if the figure were a 3D shape instead of 2D?
5. What geometric properties are essential in solving problems involving irregular polygons?

**Tip:** Always check if angles or sides are supplementary or complementary when dealing with triangle problems; this can simplify calculations significantly.

Solve this using the method to solve geometry of 2D shapes.

Learn how to solve triangle angle relationships in 2D geometry using the Law of Sines and Law of Cosines. This problem covers key concepts such as triangle properties and angle relationships, making it suitable for students in Grades 9-12.

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