https://file.aiquickdraw.com/mathbot/file/1724945228940wxlue56y.jpg

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.

Learn how to find the value of x in a circle geometry problem using the Angle at the Center Theorem. Step-by-step solution provided, analyzing angles subtended at the center and circumference. Suitable for high school students studying circle theorems.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation