1. Circle Theoram

2. Parts Parts of the Circle Circumference Major Arc radius Major Segment
Tangent
radius
Major Segment
diameter
chord
Minor Segment
Minor Arc
Major Sector
Minor Sector
Parts

3. Term’gy Introductory Terminology yo yo o o xo xo B A B B A A xo o yo
Arc AB subtends angle x at the centre.
Arc AB subtends angle y at the circumference.
Chord AB also subtends angle x at the centre.
Chord AB also subtends angle y at the circumference.

4. Th1
Theorem 1
Measure the angles at the centre and circumference and make a conjecture. xo yo o

5. The angle subtended at the centre of a circle (by an arc or chord) is twice the angle subtended at the circumference by the same arc or chord. (angle at centre)
Theorem 1
Measure the angles at the centre and circumference and make a conjecture. xo o
2xo
Angle x is subtended in the minor segment.
Watch for this one later.

6. Find the unknown angles giving reasons for your answers.B
84o xo
Example Questions 1
Find the unknown angles giving reasons for your answers. yo 2
35o
angle x =
angle y =
42o (Angle at the centre).
70o(Angle at the centre)

7. Find the unknown angles giving reasons for your answers.B
42o xo
Example Questions 3
Find the unknown angles giving reasons for your answers. po 4
62o yo qo
angle x =
angle y =
(180 – 2 x 42) = 96o (Isos triangle/angle sum triangle).
48o (Angle at the centre)
angle p =
angle q =
124o (Angle at the centre)
(180 – 124)/2 = 280 (Isos triangle/angle sum triangle).

8. Th2 The angle in a semi-circle is a right angle. Theorem 2 o o a b c d
This is just a special case of Theorem 1 and is referred to as a theorem for convenience. o
Diameter o a b c
70o d
30o e
Find the unknown angles below stating a reason.
angle a =
angle b =
angle c =
angle d =
angle e =
90o angle in a semi-circle
90o angle in a semi-circle
20o angle sum triangle
90o angle in a semi-circle
Th2
60o angle sum triangle

9. Th3 Angles subtended by an arc or chord in the same segment are equal.
Theorem 3 yo xo xo
Th3

10. Angles subtended by an arc or chord in the same segment are equal. xo yo
30o
40o
Angles subtended by an arc or chord in the same segment are equal.
Theorem 3
Find the unknown angles in each case
Angle x = angle y = 38o
Angle x = 30o
Angle y = 40o

11. Th4 The angle between a tangent and a radius is 90o. (Tan/rad)
Theorem 4 o
Th4

12. The angle between a tangent and a radius is 90o. (Tan/rad)
Theorem 4 o

13. 180 – (90 + 36) = 54o Tan/rad and angle sum of triangle.
If OT is a radius and AB is a tangent, find the unknown angles, giving reasons for your answers. T o
36o xo yo zo
30o A B
angle x =
angle y =
angle z =
180 – ( ) = 54o Tan/rad and angle sum of triangle.
90o angle in a semi-circle
60o angle sum triangle

14. The Alternate Segment Theorem.
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment.
45o xo yo
60o zo
Find the missing angles below giving reasons in each case.
angle x =
angle y =
angle z = xo yo yo xo
45o (Alt Seg)
60o (Alt Seg)
Th5
75o angle sum triangle

15. Cyclic Quadrilateral Theorem.
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o) w y
Th6 r p x z s q
Angles y + w = 180o
Angles p + r = 180o
Angles x + z = 180o
Angles q + s = 180o

16. Cyclic Quadrilateral Theorem. Theorem 6
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o)
85o
110o x y
70o
135o p r q
Find the missing angles below given reasons in each case.
angle x =
angle y =
angle p =
angle q =
angle r =
180 – 85 = 95o (cyclic quad)
180 – 135 = 45o (straight line)
180 – 110 = 70o (cyclic quad)
180 – 70 = 110o (cyclic quad)
180 – 45 = 135o (cyclic quad)

17. Th7 Two Tangent Theorem. Theorem 7
From any point outside a circle only two tangents can be drawn and they are equal in length. Q R P P T U Q R
PT = PQ T U
PT = PQ
Th7

18. Two Tangent Theorem. Theorem 7 Q O T P
From any point outside a circle only two tangents can be drawn and they are equal in length.
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. yo Q xo O
angle w =
angle x =
angle y =
angle z =
90o (tan/rad)
98o
90o (tan/rad)
49o (angle at centre)
360o – 278 = 82o (quadrilateral) zo wo T P

19. Two Tangent Theorem. Theorem 7 Q O T P
From any point outside a circle only two tangents can be drawn and they are equal in length.
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. zo Q O yo
angle w =
angle x =
angle y =
angle z =
90o (tan/rad) xo
180 – 140 = 40o (angles sum tri)
50o (isos triangle)
50o (alt seg)
80o wo
50o T P

20. Chord Bisector Theorem.
A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. O O S T
3 cm
8 cm
Find length OS
Th8
OS = 5 cm (pythag triple: 3,4,5)

21. Chord Bisector Theorem.
A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. O
Find angle x O S T
22o xo U
Angle SOT = 22o (symmetry/congruenncy)
Angle x = 180 – 112 = 68o (angle sum triangle)

22. Mixed Q 1 Mixed Questions UP R U
Mixed Questions
PTR is a tangent line to the circle at T. Find angles SUT, SOT, OTS and OST.
Angle SUT =
Angle SOT =
Angle OTS =
Angle OST =
65o (Alt seg)
130o (angle at centre)
25o (tan rad)
Mixed Q 1
25o (isos triangle)

23. Mixed Q 2 Mixed Questions
Angle w =
Angle x =
Angle y =
Angle z = O w y
48o
110o U
Mixed Questions
PR and PQ are tangents to the circle. Find the missing angles giving reasons. x z P Q R
22o (cyclic quad)
68o (tan rad)
44o (isos triangle)
Mixed Q 2
68o (alt seg)

24. Proof 1/2   Theorem 1 and 2 QED
To Prove that the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference by the same arc or chord. O C B A
Theorem 1 and 2
To prove that angle COB = 2 x angle CAB D
Extend AO to D  
AO = BO = CO (radii of same circle)
Triangle AOB is isosceles(base angles equal)
Triangle AOC is isosceles(base angles equal)
Angle AOB =  (angle sum triangle)
Angle AOC =  (angle sum triangle)
Angle COB = 360 – (AOB + AOC)(<‘s at point)
Angle COB = 360 – (  )
Angle COB = 2 + 2 = 2(+ ) = 2 x < CAB
Proof 1/2
QED

25. O
To Prove that angles subtended by an arc or chord in the same segment are equal. C B A
Theorem 3 D
To prove that angle CAB = angle BDC
With centre of circle O draw lines OB and OC.
Angle COB = 2 x angle CAB (Theorem 1).
Angle COB = 2 x angle BDC (Theorem 1).
2 x angle CAB = 2 x angle BDC
Angle CAB = angle BDC
QED
Proof 3

26. To prove that the angle between a tangent and a radius drawn to the point of contact is a right angle.
Note that this proof is given primarily for your interest and completeness. Demonstration of the proof is beyond the GCSE course but is well worth looking at. The proofs up to now have been deductive proofs. That is they start with a premise, (a statement to be proven) followed by a chain of deductive reasoning that leads to the desired conclusion.
The type of proof that follows is a little different and is known as “Reductio ad absurdum” It was first exploited with great success by ancient Greek mathematicians. The idea is to assume that the premise is not true and then apply a deductive argument that leads to an absurd or contradictory statement. The contradictory nature of the statement means that the “not true” premise is false and so the premise is proven true.
Proof 4

27. To prove that the angle between a tangent and a radius drawn to the point of contact is a right angle.
To prove that OT is perpendicular to AB
Assume that OT is not perpendicular to AB
Then there must be a point, D say, on AB such that OD is perpendicular to AB. O D C
Since ODT is a right angle then angle OTD is acute (angle sum of a triangle).
But the greater angle is opposite the greater side therefore OT is greater than OD. B T A
But OT = OC (radii of the same circle) therefore OC is also greater than OD, the part greater than the whole which is impossible.
Therefore OD is not perpendicular to AB.
By a similar argument neither is any other straight line except OT.
Therefore OT is perpendicular to AB.
Theorem 4
QED

28. To prove that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.
Theorem 5 A T B C D O
Proof 5
90 - 
To prove that angle BTD = angle TCD 2 
With centre of circle O, draw straight lines OD and OT. 
90 - 
Let angle DTB be denoted by .
Then angle DTO = 90 -  (Theorem 4 tan/rad)
Also angle TDO = 90 -  (Isos triangle)
Therefore angle TOD = 180 –(90 -  ) = 2 (angle sum triangle)
Angle TCD =  (Theorem 1 angle at the centre)
Angle BTD = angle TCD
QED

29. Proof 6     Theorem 6 alpha beta gamma delta    C
To prove that the opposite angles of a cyclic quadrilateral are supplementary (Sum to 180o).
Theorem 6    
alpha
beta
gamma
delta
To prove that angles A + C and B + D = 1800 
Draw straight lines AC and BD 
Chord DC subtends equal angles  (same segment) 
Chord AD subtends equal angles  (same segment) 
Chord AB subtends equal angles  (same segment)
Chord BC subtends equal angles  (same segment)
2( +  +  + ) = 360o (Angle sum quadrilateral)
 +  +  +  = 180o
Proof 6
Angles A + C and B + D = 1800
QED

30. To prove that the two tangents drawn from a point outside a circle are of equal length.
Theorem 7 O A B P
To prove that AP = BP.
With centre of circle at O, draw straight lines OA and OB.
OA = OB (radii of the same circle)
Angle PAO = PBO = 90o (tangent radius).
Draw straight line OP.
In triangles OBP and OAP, OA = OB and OP is common to both.
Triangles OBP and OAP are congruent (RHS)
Therefore AP = BP.
QED
Proof 7

31. To prove that a line, drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.
Theorem 8 O A B C
To prove that AB = BC.
From centre O draw straight lines OA and OC.
In triangles OAB and OCB, OC = OA (radii of same circle) and OB is common to both.
Angle OBA = angle OBC (angles on straight line)
Triangles OAB and OCB are congruent (RHS)
Therefore AB = BC
QED
Proof 8

32. Parts of the Circle
Worksheet 1

33. Th1 Worksheet 2 xo yo xo yo xo yo xo yo xo yo xo yo
Measure the angle subtended at the centre (y) and the angle subtended at the circumference (x) in each case and make a conjecture about their relationship.
Th1 xo yo o xo yo o xo yo o
Worksheet 2 xo yo o xo yo o xo yo o

34. To Prove that the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference by the same arc or chord. A O B C
Worksheet 3
Theorem 1 and 2

35. To Prove that angles subtended by an arc or chord in the same segment are equal.
Worksheet 4
Theorem 3

36. To prove that the angle between a tangent and a radius drawn to the point of contact is a right angle. O B T A
Worksheet 5
Theorem 4

37. To prove that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. D B O C T A
Worksheet 6
Theorem 5

38. Worksheet 7     Theorem 6 Alpha Beta Chi delta
To prove that the opposite angles of a cyclic quadrilateral are supplementary (Sum to 180o). B A C D
Worksheet 7    
Theorem 6
Alpha
Beta
Chi
delta

39. Worksheet 8 A O P B Theorem 7
To prove that the two tangents drawn from a point outside a circle are of equal length. A O P B
Worksheet 8
Theorem 7

40. To prove that a line, drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord. O A C B
Worksheet 9
Theorem 8