0580

Circle Theorems

Geometry

The angle at the centre is twice the angle at the circumference

  • Take a pair of points on the circumference and join each of them to the centre, and also to some third point on the circumference
  • Whatever angle appears at the centre is double the one appearing at that third point
  • The shape often looks like an arrowhead, and the two angles must be subtended by the same arc
The same theorem in a less obvious arrangement: P and Q are joined to the centre and to a third point close to P, so the shape no longer looks like an arrowhead, but the angle at the centre is still twice the angle at the circumference
Source: Angles at centre & circumference by Save My Exams
  • Where the centre angle is reflex, it is that reflex value which doubles the circumference angle, not the smaller one beside it
A circle where the third point sits on the short arc, so the angle at the centre facing the same arc is the reflex one, marked 2x, and it is that reflex angle rather than the smaller one beside it which is double the angle x at the circumference
Source: Angles at centre & circumference by Save My Exams
  • An angle of 33° at the circumference gives 66° at the centre
A circle with P and Q on the circumference and the arc between them picked out in red: both are joined to the centre, giving an angle of 2x there, and both are joined to a third point on the circumference, where the angle is x
Source: Angles at centre & circumference by Save My Exams

Angles in the same segment are equal

  • Two points on the circumference joined to two different third points give two equal angles
  • Both angles must start from the same pair of points, and both third points must lie on the same side of the chord joining them
  • Drawing in that chord makes the segment visible and settles whether the two angles really match
P and Q on a circle, each joined to two different points on the circumference above them, giving two equal angles marked y; the same two points are also joined to a second pair below, giving two more equal angles marked x
Source: Circles & segments by Save My Exams

Opposite angles in a cyclic quadrilateral add to 180°

  • A cyclic quadrilateral has all four of its corners on the circumference
  • Its two pairs of opposite angles each total 180°
  • An angle of 104° therefore sits opposite one of 76°
  • Check all four corners are on the circle before using it, since a quadrilateral with only three on the circumference does not qualify
A quadrilateral ABCD drawn inside a circle with centre O, every one of its four corners sitting on the circumference, which is what makes it cyclic
Source: Cyclic quadrilaterals by Save My Exams

The alternate segment theorem

  • The angle between a tangent and a chord equals the angle in the alternate segment, that is the angle at the circumference on the other side of that chord
  • Look for a triangle with all three corners on the circle and one corner touching a tangent
  • Find the side that forms the tangent angle, and the matching angle is the one facing it from across the triangle
  • An angle of 58° between tangent and chord matches one of 58° in the alternate segment
  • The name is the reason, so write "alternate segment theorem" in full
A triangle with all three corners on a circle, one corner resting on a tangent: the angle between the tangent and one side of the triangle is marked in red, and so is the equal angle facing it from the opposite corner of the triangle, on the far side of that chord
Source: Alternate segment theorem by Save My Exams
Worked example

Two of the Extended theorems

A, B and C lie on a circle with centre O. Angle AOC at the centre is 130°, and B is on the major arc. (a) Find angle ABC. (b) ABCD is a cyclic quadrilateral and angle BAD is 85°. Find angle BCD.

A circle with centre O marked. Points A, B and C lie on the circumference, with B on the major arc AC. The radii OA and OC are drawn and the angle between them at O, on the side away from B, is marked 130 degrees. The chords AB and BC are drawn, and the angle at B is not labelled. The figure is marked NOT TO SCALE.

Solution:

  • (a) Angle AOC at the centre and angle ABC at the circumference both stand on the same arc AC
  • The angle at the centre is twice the angle at the circumference
  • Angle ABC = 130 ÷ 2 = 65°
  • The reason is "the angle at the centre is twice the angle at the circumference"
  • (b) BAD and BCD are opposite angles of a cyclic quadrilateral, so they add to 180°
  • Angle BCD = 180 − 85 = 95°
  • The reason is "opposite angles of a cyclic quadrilateral add to 180°"
  • Check that all four corners really sit on the circumference before using that second theorem — a quadrilateral with one vertex at the centre is not cyclic