1. Remember to look for “basics”
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
tangent
Diameter
radius
chord

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

6. LOOKS DIFFERENT BUT STILL THE CENTRE
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. PALE BLUE AREA IS THE SEGMENT
RULE 3: Angles at the circumference in the same SEGMENT of a circle are equal

10. PALE BLUE AREA IS THE SEGMENT
RULE 3: Angles at the circumference in the same SEGMENT of a circle are equal

11. THEOREM 3: 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….)

12. THEOREM 4: Opposite angles sum to 180 in a cyclic quadrilateral
CYCLIC QUADRILATEARAL
MUST touch the circumference at all four vertices
890
1100
700
910
THEOREM 4: Opposite angles sum to 180 in a cyclic quadrilateral

13. Enter the world of tangents and chords…..
A tangent is a line that touches a circle at one point only. This point is called the point of contact.
A chord is a line that joins two points on the circumference.
tangent
chord

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

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

16. Theorem 7 – The line joining an external point to the centre of a circle bisects the angle between the tangents
350
700
350

17. Theorem 5&7 – combined can help you find the missing angles…..
900 x
350
700
350 y
900

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

19. 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

20. Theorem 9 – Alternate angle theorem

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

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

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

24. Hope this has been useful – if you are still stuck, double check your “basics” from slide 2.