The angle subtended by an arc at the centre is ........ the angle subtended by it at any point on the remaining part of the circle.

Question

Accepted Answer

The correct answer is Option 1: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

### Explanation:

1. **Why Option 1 is Correct:**
   - In a circle, there is a well-known theorem called the "Angle at the Centre Theorem" or "Central Angle Theorem." This theorem states that the angle subtended by an arc at the centre of the circle is exactly twice the angle subtended by the same arc at any point on the remaining part of the circle. This is a fundamental property of circles and is used extensively in geometry.
   - For example, if an arc subtends an angle of 30 degrees at any point on the circle, it will subtend an angle of 60 degrees at the centre of the circle. This relationship holds true for any arc and any point on the circle.

2. **Why the Other Options are Incorrect:**
   - **Option 2 (Triple):** This option suggests that the angle at the centre is three times the angle at the circumference. However, this contradicts the Angle at the Centre Theorem, which clearly states that the angle at the centre is only double, not triple.
   - **Option 3 (Same):** This option implies that the angle at the centre and the angle at the circumference are equal. This is incorrect because, according to the theorem, the central angle is always twice the angle at the circumference, not the same.
   - **Option 4 (Four times):** This option suggests that the angle at the centre is four times the angle at the circumference. This is incorrect as per the Angle at the Centre Theorem, which specifies that the central angle is only twice the angle at the circumference, not four times.

In conclusion, Option 1 is the correct answer because it aligns with the established geometric theorem regarding angles subtended by arcs in a circle. The other options do not conform to this theorem and are therefore incorrect.