1. Circles

2. Radius
Diameter

3. Tangent
(major) Segment
Chord
(minor) Segment
Arc

4. Circumference of a circle =or r D p 2

5. Area of a circle =
r2

8. Angles held up by the diameter are called “Angles in the semi-circle” and are all 900

9. Isosceles triangles are formed by two radii. .
The angle in a semicircle is 90°
Isosceles triangles are formed by two radii . .
Chord
Any chord bisector is a diameter
Radius
Tangent
Tangent and Radius meet at 90°
90°

10. Angles in the same segment are equalx

11. x 2x
Angles from chord to centre are twice the size of angles from chord to circumference

12. The angle at the centre is 1800 , so the angle at the circumference is half that - 900

13. Opposite angles in cyclic quadrilateral add up to 1800 (supplementary)
680 . c a b
= 1120 opposite angle of a cyclic quadrilateral

14. Opposite angles in cyclic quadrilateral add up to 1800 (supplementary)
680 . c a b
= 1120 opposite angle of a cyclic quadrilateral
Adjacent angles in cyclic trapezium are equal - angles subtended by an arc.

15. a
420 b . d c
Find the missing angles a, b, c and d

16. . d = 840 angle at the centre is twice the angle at the circumference
= 420 angle in the same segment
420 b
= 420 angle in the same segment . d c

17. 770 . c a b
Find the missing angles a, b, and c

18. . c a = 770 adjacent angle of a cyclic trapezium b
= 1030 adjacent angle of a cyclic trapezium, or opposite angle of a cyclic quadrilateral b
= 1030 opposite angle of a cyclic quadrilateral

19. For the following circles, where O is the centre of the circle, find the missing angles
870
1350 c . . . e o o d o a b
470
480 i k
920 . . f
580 h o l o j
390 g m
1100

20. For the following circles, where O is the centre of the circle, find the missing angles
870
1350 . . .
d = 900
e = 960 d o o o e a
a =930
b =450 b
470
480
k=320 f
h=1220
i=900
920 .
f = 390 .
580 i k h l o j o
l=460
j=320
390
g = 310 m g
m=460
310
1100

21. Two tangents drawn from an outside point are always equal in length,
so creating an isosceles situation with two congruent right-angled triangles

22. Reminder of “segment”
(major) Segment
(minor) Segment

23. The angle in the opposite segment
The angle between chord and tangent
The angle between a chord and a tangent = the angle in the opposite segment

24. Two tangents drawn from an outside point are always equal in length,
so creating an isosceles situation with two congruent right-angled triangles
The angle in the opposite segment
The angle between chord and tangent
The angle between a chord and a tangent = the angle in the opposite segment m m

25. Angle between tangent and radius is a right angle
900 B
Angle at the centre is twice the angle at the circumference
600 A
850
1700
100 O
250
The angle between a chord and a tangent = the angle in the opposite segment
950 C E
250
In kite BEDO, BED = 360-known angles =100 D
Opposite angles of a cyclic quad are supplementary
Find each of the following angles
OBE BOD BED BCD CAB