One arc, one angle

The idea

An arc subtends the same angle at every point of the circle outside the arc, so angles in the same segment of a circle are all equal.

Walk round the circle

An arc AB looks the same size from the centre, every time. Now stand on the circle itself and look at the arc AB from different places on the far side.

You walk round the circle, always outside the arc AB. What happens to the angle ∠ADB?

• It stays the same
• It gets bigger
• It gets smaller

Place D

A and B are fixed. The orange arc AXB subtends 100° at the centre. Touch anywhere above AB to place D, and read ∠ADB.

All equal

An arc subtends the same angle at every point of the circle outside the arc. So angles in the same segment of a circle are equal.

Each of these angles is half the angle the arc subtends at the centre, so they are all equal. A point not on the circle, inside it or outside it, sees the arc at a different angle.

Notes

An arc subtends the same angle at every point of the circle outside the arc: angles in the same segment are equal.

Check yourself

D and E are points of a circle, both outside the arc AB. ∠ADB = 45°. What is ∠AEB, in degrees?

Answer: 45 °

Angles in the same segment are equal, so ∠AEB = ∠ADB = 45°.

P, Q and R are points of a circle, all outside the arc AB. How do ∠APB, ∠AQB and ∠ARB compare?

D is a point inside the circle, on the same side of AB as the points of the circle outside the arc AB. How does ∠ADB compare with the angle the arc subtends at points of the circle?

An arc AB subtends 100° at the centre. D and E are points of the circle outside the arc. What is ∠ADB + ∠AEB, in degrees?

Answer: 100 °

Each angle is half of 100°, so ∠ADB = ∠AEB = 50°. Their sum is 50° + 50° = 100°.

• All three are equal — correct. Yes. An arc subtends the same angle at every point of the circle outside the arc.
• The one nearest to AB is the biggest. The distance from AB does not matter, as long as the point is on the circle.
• The one farthest from AB is the biggest. The distance from AB does not matter, as long as the point is on the circle.
• It is smaller. It is the other way round: inside the circle the angle is bigger, and outside the circle it is smaller.
• It is bigger — correct. Yes. Only points on the circle give the same angle. From a point inside the circle you see the arc at a bigger angle.
• It is the same. Only points ON the circle give the same angle.