https://file.aiquickdraw.com/mathbot/file/17260221312686hw3wjjp.jpg

Let's analyze the image step by step and solve the two questions:

### Diagram Breakdown:
- The figure shows a circle with a triangle \( ABE \) inscribed in it, and an external tangent line that extends at point \( B \) to points \( C \) and \( D \).
- The angle at the center of the circle subtended by arc \( AE \) is \( 280^\circ \).
- Angle \( x \) is formed by the tangent at point \( B \) and line \( BD \).
- Line \( CD \) appears to be the tangent, with point \( C \) and \( D \) lying outside the circle.

### Part i) Find \( \angle x \) with reason:
- **Key Concept**: The angle between a tangent and a chord through the point of contact (e.g., \( \angle x \)) is equal to the angle subtended by the chord in the alternate segment.
- Here, \( \angle ABE \) subtends arc \( AE \). Since the angle at the center is \( 280^\circ \), the angle subtended by arc \( AE \) at the circumference (i.e., \( \angle ABE \)) is half of that:
  \[
  \angle ABE = \frac{280^\circ}{2} = 140^\circ
  \]
- Using the alternate segment theorem, \( \angle x \) is equal to \( \angle ABE \), so:
  \[
  \boxed{x = 140^\circ}
  \]

### Part ii) Find \( \angle DBE \) with reason:
- \( \angle DBE \) is an external angle between the lines \( BD \) and \( BE \). Since the tangent \( BD \) is perpendicular to the radius at \( B \), \( \angle DBE \) is \( 90^\circ \).

Thus:
\[
\boxed{\angle DBE = 90^\circ}
\]

Would you like further clarification or details?

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Here are five related questions for further understanding:
1. How do we determine the angle at the center of a circle subtended by a chord?
2. What is the alternate segment theorem in circle geometry?
3. Can the tangent-segment relationship be used to find angles in other scenarios?
4. How does the angle subtended by a diameter at the circumference behave?
5. How can inscribed angles be used to find unknown angles in a circle?

**Tip**: In geometry, always look for relationships between angles formed by tangents, chords, and radii, especially using theorems like the alternate segment theorem or the tangent-radius property.

Refer to the diagram below: i) Find angle x with reason. ii) Find angle DBE with reason.

Solve the circle geometry problem where a triangle is inscribed in a circle, and a tangent extends from point B. Learn how to apply the alternate segment theorem and the tangent-radius property to find angles.

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