Angle at centre = 2 × angle at circumference
Same arc, same side. The centre angle is double.
If two angles in a circle subtend the same arc — one at the centre and one at the circumference (on the same side) — the angle at the centre is twice the angle at the circumference.
Worked. Centre angle = . Find the circumference angle subtending the same arc.
- .
- Reason: angle at centre is twice angle at circumference.
Special case. If the chord is a DIAMETER, the centre angle is , so the circumference angle is — that's the angle in a semicircle = theorem.
- Same arc, same side.
- Centre angle = circumference angle.
- Diameter case: angle in semicircle = .
- ALWAYS state the theorem name in the reason.