What is the relationship between the angle at the centre and the angle at the circumference?

The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

In more detail, this relationship is a fundamental property of circles in geometry. When you have a circle, and you draw two lines from the centre to the circumference, these lines form an angle at the centre. If you then draw two lines from the same points on the circumference to another point on the circumference, this forms an angle at the circumference. The angle at the centre is always twice as large as the angle at the circumference when both angles are subtended by the same arc.

To visualise this, imagine a circle with a central point O. Choose two points A and B on the circumference and draw lines OA and OB. The angle ∠AOB is the angle at the centre. Now, pick another point C on the circumference and draw lines AC and BC. The angle ∠ACB is the angle at the circumference. According to the circle theorems, ∠AOB = 2∠ACB.

This property is very useful in solving various problems related to circles, such as finding missing angles or proving that certain lines are parallel or perpendicular. It also helps in understanding the properties of cyclic quadrilaterals, where the opposite angles add up to 180 degrees. Remembering this relationship can make many geometry problems much easier to solve.