1. CIRCLE THEOREMS
Questions normally ask for the size of an angle and reasons why.
In each case use the theorems you know and what is special.
Any triangles made from radii will always be isosceles.

2. Tangent and Radius are perpendicular
CIRCLE THEOREMS
Tangent and Radius are perpendicular

3. Angle in a Semi-Circle is always 90°
CIRCLE THEOREMS
Angle in a Semi-Circle is always 90°

4. The angle at the Centre is twice that at the circumference
CIRCLE THEOREMS
The angle at the Centre is twice that at the circumference

5. Angles from the same chord are equal
CIRCLE THEOREMS
Angles from the same chord are equal
Angles A and C are equal from the chord BD
In the same way angle B and D are equal, coming from the chord AC

6. The opposite angles in a cyclic quadrilateral add to 180°
CIRCLE THEOREMS
The opposite angles in a cyclic quadrilateral add to 180°
A and C add to 180
and so do B and D

7. Alternate Segment theorem
CIRCLE THEOREMS
Alternate Segment theorem
Angle between the tangent and the chord (angle a on the diagram) equals the angle in the alternate(other) segment – angle b

8. Examples
Try the following examples. The question is on one page, go to next page for a solution

9. CIRCLE THEOREMS
Examples

10. CIRCLE THEOREMS Examples In Δ OAB angles A and B are equal
OA=OB as radii which makes isosceles
Angle OBA will then be (180-2x) ÷ 2 which is (90-x)
OB and OT are perpendicular as they are radius and tangent and so Angle ABT will be 90-(90-x) which is x

11. CIRCLE THEOREMS
Examples

12. CIRCLE THEOREMS Examples
Draw the line AO and label angles at B and C as b and c
In Δ OAC,
OA=OC as radii so Δ is isosceles
Angle OAC = OCA = c
Angle AOC is 180 – 2c
In same way in ΔOBC angle BOC is 180 – 2b
Angles around a point are 360 so angle BOC=360- AOC – BOC
= 360 – (180 -2b) –(180-2c)
= 2b + 2c
= 2 x ( b + c) = 2 x angle CAB b c

13. CIRCLE THEOREMS Examples (a) Work out the size of angle AOD°
The diagram shows a circle centre O. A, B and C are points on the circumference.
DCO is a straight line.
DA is a tangent to the circle.
Angle ADO = 36°
(a) Work out the size of angle AOD°
(b) Work out the size of angle ABC.
Give reasons for your answers

14. CIRCLE THEOREMS Angle OAD is 90 as OA is a radius and AD a tangent
Angle AOD is 54° as OAD is a triangle and OAD is 90. Angle ABC is the angle at the circumference when AOC is at the centre and so is 27°
Angle at centre theorem

15. CIRCLE THEOREMS
Examples

16. CIRCLE THEOREMS Angle PQT = angle PTA = 58° Alternate segment theorem
In Δ PQT, PQ=QT and so it isosceles and the angles at P and T are equal and as PQT is 58, they are (180-58) ÷ 2 which is 61
Angle POT is = 32 because angle OAT is 90 (radius/tangent)
Finally OTQ = = 29°

17. CIRCLE THEOREMS
Examples

18. CIRCLE THEOREMS
AD is a chord and angles ABD and ACD are from that chord and so are equal.
 ACD = 54°

19. CIRCLE THEOREMS
Examples

20. CIRCLE THEOREMS ABCD is a cyclic quadrilateral and so angle BAD = 50
Angle BOD is the angle at the centre and so is 100°
In the quadrilateral OBDC we now have 100 and 130 which leaves = 130 for the other two.
As OB=OD (radii), OBDC is symmetric and angle OBC and ODC are equal and must be 130 ÷ 2 = 65°

21. CIRCLE THEOREMS
Examples

22. CIRCLE THEOREMS ABCD is a cyclic quadrilateral and so angle D is 52°
Angle AOC is the angle at the centre and so will be twice that at the circumference and so is 2 x 52 = 104°

23. CIRCLE THEOREMS
Examples

24. CIRCLE THEOREMS As no angles are known set the one you want to x
BC=BD so Δ BCD is isosceles and so BCD is also x
Angle C is 90 as it is the angle in a semi-circle and so angle ACD is 90 - x
Because AC and BD are parallel, angles ACD and CDB are equal (alternate angles) and so x = 90 – x
Solving gives x = 45°