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In the diagram you provided, a circle is shown with various labeled angles, and we need to find the values of \(x\).

### Observations and Angle Relationships
1. **Angles on a Straight Line**: The angles along line \(BD\) must sum to 180°.
2. **Angles at a Point**: The total angle around point \(A\) is 360°.

### Calculating \(x\)
- Angles along line \(BD\):
  \[
  (x + 20°) + (2x + 40°) = 180°
  \]
  Combining like terms:
  \[
  3x + 60° = 180°
  \]
  Solving for \(x\):
  \[
  3x = 120°
  \]
  \[
  x = 40°
  \]

The angle \(x\) measures 40°. If you need further clarification on any steps or have additional questions, please let me know!

### Related Questions
1. How can the properties of circle geometry (e.g., angles subtended by the same arc) be used to solve other problems?
2. What is the measure of angle \(ACB\) in this circle?
3. How do we calculate the angles formed by intersecting chords in a circle?
4. If \(AB\) and \(CD\) are diameters, what would be the measure of angle \(ADB\)?
5. How does the Inscribed Angle Theorem apply to quadrilaterals inscribed in a circle?

### Tip
Always check if multiple angle rules or theorems apply in a complex diagram to ensure you've considered all relationships between the angles and segments.

In the diagram, find the value of x using angle relationships in a circle.

Learn how to solve for x in a circle involving intersecting chords and the relationships between angles. This problem applies core geometric theorems such as the Inscribed Angle Theorem and properties of angles on a straight line. Suitable for students in Grades 8-10.

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