Four corners on one circle, and the pairs that total 180°
Join four points in order round one circle. The shape gets a special name.
- Four corners, A, B, C, D, in order on a circle.
- Chord BD, with angle A standing on one side.
- Angle C, standing on the other side of BD.
- Angle A plus angle C, marked as 180°.
The rule
Put all four corners of a quadrilateral ABCD on one circle. It becomes a cyclic quadrilateral. Angle A stands on chord BD, facing one arc. Angle C stands on the same chord, facing the other arc. Each angle is half the centre angle on its arc. The two centre angles total 360°, so the two halves total 180°. So angle A plus angle C is 180°. Use the same argument on chord AC. Angle B plus angle D also equals 180°. Opposite angles of a cyclic quadrilateral total 180°.
Worked, and checked
Take a cyclic quadrilateral ABCD. Its angles in order are 85°, 110°, 95° and 70°.
- A + C: 85 + 95 = 180.
- B + D: 110 + 70 = 180.
- All four total 360°, as any quadrilateral does.
| Claim | Any quadrilateral | Cyclic one |
|---|---|---|
| Four angles total 360° | Yes | Yes |
| Opposite angles total 180° | Not always | Always |
Watch out!
A total of 360° does not make both opposite pairs 180°. Try 150°, 30°, 150° and 30°, in order. The pairs come to 300° and 60°, not 180°.
Slide corner C. Watch the two pairs.
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