To solve for $x$ in the given circle diagram:

Step 1: Analyze the Diagram

• The circle has points $A$, $B$, $C$, and $D$ on its circumference.
• The center of the circle is labeled $O$.
• Angle $\angle BOD = x^\circ$.
• $\angle BAC = 95^\circ$ and $\angle ABC = 30^\circ$.

Step 2: Understand the Relationships

• Angle at the center ( $\angle BOD$ ): The angle subtended by arc $BD$ at the center is twice the angle subtended by the same arc at the circumference ( $\angle BAC$ ).

Step 3: Apply the Angle at Center Theorem

• According to the angle at the center theorem: $\angle BOD = 2 \times \angle BAC$ Substituting the given values: $x = 2 \times 95^\circ$ $x = 190^\circ$

Thus, $x = 190^\circ$.

Conclusion

The value of $x$ is $190^\circ$.

Would you like further details or have any questions? Here are some related questions to explore:

1. What is the significance of the angle at the center theorem in circle geometry?
2. How would you calculate the angle if point $C$ were placed differently on the circle?
3. Can the angle $\angle BAC$ be greater than $180^\circ$? Why or why not?
4. What is the relationship between opposite angles in a cyclic quadrilateral?
5. How does the concept of supplementary angles apply to angles subtended by the same arc?

Tip: Remember that in circle geometry, angles subtended by the same arc at different points on the circumference are always equal.