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To solve for \( x \) in this geometry problem, we need to analyze the diagram and information given. The circle has its center at \( O \), and lines \( AC \) and \( BOD \) are straight. Let's use circle theorems to approach the solution.

### Key observations:
1. **Line \( BOD \) is a straight line**: This tells us that \( \angle BOD = 180^\circ \) because it's a straight angle.
2. **Opposite angles in a cyclic quadrilateral**: Since points \( A \), \( B \), \( C \), and \( D \) lie on the circle, quadrilateral \( ABCD \) is cyclic. For any cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \).

Let's denote the angles in the diagram:
- \( \angle AOC = x \)
- \( \angle BOD = 180^\circ - 2x \) because they are supplementary (on a straight line \( BOD \)).

From the cyclic quadrilateral property:
\[
\angle DAB + \angle DCB = 180^\circ
\]

Using this information, we can now proceed with calculating \( x \). I will compute the necessary steps.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the solution manually if you'd like. Would you like me to explain the steps in more detail?

Here are some related questions that might help you further:
1. What is a cyclic quadrilateral, and why do opposite angles sum up to \( 180^\circ \)?
2. How does the straight line property of \( BOD \) influence the angle \( x \)?
3. Can the angle subtended by a diameter always equal \( 90^\circ \), and why?
4. What other theorems about circles involve angles and chords?
5. How can you find unknown angles in a circle without knowing all sides or angles?

**Tip**: Remember that understanding the properties of cyclic quadrilaterals and straight lines is key in solving most circle geometry problems.

The diagram shows a circle with center O. AC and BOD are straight lines. Find the value of x.

Learn how to solve for x in a circle with center O, where AC and BOD are straight lines. This geometry problem involves key circle theorems such as the cyclic quadrilateral and the angle at the center. Suitable for students in Grades 9-11.

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