1. Draw and label on a circle:
Centre
Radius
Diameter
Circumference
Chord
Tangent
Arc
Sector (major/minor)
Segment (major/minor)

2. Circle Fact 1. Isosceles Triangle
Any triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles. r a a r a

3. Circle Fact 2. Tangent and Radius
The tangent to a circle is perpendicular to the radius at the point of contact.

4. Circle Fact 3. Two Tangents
The triangle produced by two crossing tangents is isosceles.

5. Circle Fact 4. Chords
If a radius bisects a chord, it does so at right angles, and if a radius cuts a chord at right angles, it bisects it.

6. Circle Theorem 1: Double Angle
The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

7. Circle Theorem 2: Semicircle
The angle in a semicircle is a right angle.

8. Circle Theorem 3: Segment Angles
Angles in the same segment are equal.

9. Circle Theorem 4: Cyclic Quadrilateral
The sum of the opposite angles of a cyclic quadrilateral is 180o.

10. Circle Theorem 5: Alternate Segment
The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

11. Circle Theorem 1: Double Angle
Proof
The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

12. b b
2a + 2b
= 2(a + b) b
180 – 2a a a

13. Circle Theorem 2: Semicircle
Proof
The angle in a semicircle is a right angle.

14. b
360 – 2(a + b) = 180
180 = 2(a + b)
180 – 2b
90 = (a + b)
180 – 2a b a a

15. Circle Theorem 3: Segment Angles
Proof
Angles in the same segment are equal.

16. a a 2a

17. Circle Theorem 4: Cyclic Quadrilateral
Proof
The sum of the opposite angles of a cyclic quadrilateral is 180o.

18. a
2a + 2b = 360
2(a + b) = 360 2b
a + b = 180 2a b

19. Circle Theorem 5: Alternate Segment
Proof
The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

20. 90 - a a
180 – 2(90 – a) 2a
180 – a
90 - a a

21. 1 2 3 4 5 Double Angle Semicircle Segment Angles Cyclic Quadrilateral
Alternate Segment