The document outlines various concepts and theorems related to circles, including definitions of terms such as radius, diameter, and tangent. Key circle theorems are detailed, including the properties of angles formed in semicircles, cyclic quadrilaterals, and chords. Important facts and relationships about isosceles triangles and tangents to circles are highlighted.

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

2. a
a
180 - 2a
r
r
Circle Fact 1. Isosceles Triangle
Any triangle AOB
with A & B on the
circumference
and O at the
centre of a circle
is isosceles.

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
The angle
subtended by an
arc at the centre
of a circle is twice
the angle
subtended at the
circumference.

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

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

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

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

16. a
a
2a

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

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

19. 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.

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

21. Double Angle Semicircle
Segment Angles Cyclic Quadrilateral Alternate Segment