This document summarizes 10 theorems related to circles: 1. The angle at the center of a circle is twice the angle at the circumference subtended by the same arc. 2. Every angle at the circumference of a semicircle that is subtended by the diameter is a right angle. 3. Opposite angles sum to 180 degrees in a cyclic quadrilateral where the sides touch the circle. 4. Angles in the same segment of a circle are equal. 5. A tangent is perpendicular to the radius at the point of contact. 6. Tangents from the same external point are equal in length. 7. A line joining an external

1. Circle Theorem
Remember to look for “basics”
•Angles in a triangle sum to 1800
•Angles on a line sum to 1800
•Isosceles triangles (radius)
•Angles about a point sum to 3600

2. Name parts of a circle

3. MUST BE
THE
CENTER
400
800
THEOREM 1: ANGLE at the CENTRE
of the CIRCLE is twice the angle at the
circumference subtended by the same
arc

4. This rule can be hard to spot…..

5. MUST BE
THE
1150 CENTER
2300
THIS IS THE ONE MOST PEOPLE DON’T SEE......

6. 800
400
LOOKS
DIFFERENT
BUT STILL
THE CENTRE

7. 900
1800
SPECIAL CASE OF THE SAME RULE……… BUT
MAKES A RULE IN ITS OWN RIGHT!!

8. 900
THEOREM 2: Every angle at the circumference
of a SEMICIRCLE, that is subtended by the
diameter of the semi-circle is a right angle.

9. CYCLIC
QUADRILATEARAL
MUST touch the
circumference at
all four vertices
890
1100
700
910
THEOREM 3: Opposite angles sum to 180 in a
cyclic quadrilateral

10. Now have a go at Worksheet 1 

11. PALE BLUE
AREA IS THE
SEGMENT
RULE 4: Angles at the circumference in the
same SEGMENT of a circle are equal

12. PALE BLUE
AREA IS THE
SEGMENT
RULE 4: Angles at the circumference in the
same SEGMENT of a circle are equal

13. THEOREM 4: Angles at the circumference in
the same SEGMENT of a circle are equal
NOTE: Will lead you to SIMILAR triangles (one
is an enlargement of the other….)

14. • A tangentthe line thatof tangents andone
Enter is a world touches a circle at
point only. This chords….. the point of
point is called
contact.
• A chord is a line that joins two points on the
circumference.

15. Theorem 5 – A tangent is
perpendicular to a radius
900
radius

16. Theorem 6 – Tangents to a circle from
the same point are equal in length

17. Theorem 7 – The line joining an external point to the
centre of a circle bisects the angle between the tangents
350 0
70
350

18. Theorem 5&7 – combined can help you find the missing
angles…..
900
x 350 0
70
y 350
900

19. Theorem 8 – A radius bisects a chord at 900
MIDPOINT
OF THE
CHORD
chord
900
And the chord will be cut perfectly in half!!!

20. Have a go at worksheet 2 

21. Theorem 9 – Alternate angle theorem
Need a tangent
And a triangle that joins the tangent and two other
points on the circumference of the circle

22. Theorem 9 – Alternate angle theorem

23. Theorem 9 – Alternate angle theorem
The angle between a tangent and a chord
Is equal to the angle in the alternate segment

24. Theorem 9 – Alternate angle theorem
The angle between a tangent and a chord
Is equal to the angle in the alternate segment

25. COMMON EXAM ERROR!
IT TOONLY A
IS IS THINK
DIAMETER
THIS IS A
IF YOU ARE
DIAMETER –
TOLD SO…
SO..
THIS MUST
READ
BE 900 –
QUSETIONS
“TANGENT
CAREFULLY..
MEETS
RADIUS”