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The image shows a circle with points \( A \), \( B \), \( C \), and \( D \) on the circumference, and point \( O \) as the center. The angle \( \angle AOC = 140^\circ \), and the angle \( \angle OBD = 2x \). 

### To find the missing values:
1. **Relation between angles at the center and on the circumference**:
   - The angle at the center (e.g., \( \angle AOC \)) is twice the angle at the circumference subtended by the same arc (arc \( AC \)).
   - Therefore, \( \angle ADC \) is half of \( \angle AOC \).
   - \( \angle ADC = \frac{140^\circ}{2} = 70^\circ \).

2. **Angle relationships** in cyclic quadrilateral \( ABCD \):
   - Since \( A \), \( B \), \( C \), and \( D \) are points on a circle, \( \angle A + \angle C = 180^\circ \) (opposite angles of a cyclic quadrilateral).
   - You can use this information to further solve the unknown angles.

### Do you want more detailed calculations or help with specific parts of this problem?

Here are 5 related questions:
1. How does the angle subtended by an arc at the center compare to the angle at the circumference?
2. What is the significance of a cyclic quadrilateral in geometry?
3. How can we calculate angle relationships in a cyclic quadrilateral?
4. What are the properties of angles in a circle?
5. How can you find the value of \( x \) in the diagram?

**Tip**: Always remember the relationship between angles subtended by the same arc at the center and on the circumference.

The diagram shows a circle with points A, B, C, and D on the circumference and point O as the center. The angle ∠AOC is 140°, and ∠OBD is 2x. Find the value of the unknown angles.

This problem involves finding the unknown angles in a circle with a cyclic quadrilateral and applying key theorems like the angle at the center being twice the angle at the circumference and opposite angles of a cyclic quadrilateral summing to 180°. Suitable for students in Grades 8-10.

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