Understanding the Angles in a Circle
To prove that the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle, we can use the concept of inscribed angles.
Definitions
- Central Angle: The angle formed at the center of the circle by two radii that connect the center to the endpoints of an arc.
- Inscribed Angle: The angle formed by two chords in a circle which share an endpoint. This endpoint is the vertex of the angle.
Proof
1. Draw a Circle: Consider a circle with center O, and let A and B be the endpoints of an arc.
2. Identify Points: Let C be any point on the remaining part of the circle.
3. Draw Angles:
- The central angle AOB is formed at the center O.
- The inscribed angle ACB is formed at point C.
4. Triangles and Properties:
- Triangles OAC and OBC are created by drawing lines from O to A, O to B, and the line segment AB.
- The angles ∠OAC and ∠OBC are equal to the inscribed angle ACB.
5. Angle Relationships:
- In triangle OAC, the angle AOB is equal to the sum of angles OAC and OCA.
- By the properties of circles, it can be established that the central angle AOB is twice the inscribed angle ACB.
Conclusion
Thus, it is proven that the angle subtended by an arc at the center of the circle (AOB) is indeed double the angle subtended by the same arc at any point on the remaining part of the circle (ACB). This fundamental property is crucial in understanding circle geometry and its applications in various fields.

It is given in the theorem portion of the ncert book.