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1 PLC Papers Created For: ed by use of accompanying mark schemes towards the rear to attain 8 out of 10 marks over time by completing

2 Circle Theorems 1 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A, B, C and D are points on the circumference of a circle with centre O. Angle ABC = 116 Find the size of the angle marked x. Give reasons for your answer. (Total 3 marks)

3 Question 2 Diagram NOT accurately drawn B, C and D are points on the circumference of a circle, centre O. AB and AD are tangents to the circle. Angle DAB = 50 Work out the size of angle BCD. Give a reason for each stage in your working. (Total 4 marks)

4 Question 3 B and C are points on the circumference of a circle, centre O. AB and AC are tangents to the circle. Angle BAC = 40. Find the size of angle BCO. (Total 3 marks) Total /10

5 Circle Theorems 2 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A and B are points on the circumference of a circle, centre O. AC and BC are tangents to the circle. Angle ACB = 36. Find the size of angle OBA. Give reasons for your answer. (Total 3 marks)

6 Question 2 * A, B, C and D are points on the circumference of a circle, centre O. Angle AOC = y. Find the size of angle ABC in terms of y. Give a reason for each stage of your working. (Total 4 marks)

7 Question 3 B, C and D are points on the circumference of a circle, centre O. ABE and ADF are tangents to the circle. Angle DAB = 40 Angle CBE = 75 Work out the size of angle ODC. (Total 3 marks) Total /10

8 Circle Theorems 3 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 ADB is the tangent at D to the circle, centre O. C is a point on the circumference. Angle OCD = 25. Calculate angle CDA. Show each step of your calculation. Not to scale... [3]

9 Question 2 B, D and E are points on a circle, centre O. DE is a diameter. AC is the tangent to the circle at B. Angle BDE = 18. Not to scale (a) Give the reason why angle EBD is [1] (b) Calculate angle ABD. Give reasons for your answer. Angle ABD =... because [2]

10 Question 3 A, B and C are points on a circle, centre O. TCD is a tangent to the circle. Angle BAC = 52. Not to scale Find angles BOC and BCT, giving your reasons. Angle BOC =... because [2] Angle BCT =... because [2] Total / 10

11 Total marks / 10

12 Circle Theorems 4 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 (b) Work out the value of angle BAX. You must show all of your working. (5)

13 Question 2 Write down a value for the following angles, and give a reason. (2) (2) Total / 10

14 PLC Papers Created For: ed by use of accompanying mark schemes towards the rear to attain 8 out of 10 marks over time by completing

15 Circle Theorems 1 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A, B, C and D are points on the circumference of a circle with centre O. Angle ABC = 116 Find the size of the angle marked x. Give reasons for your answer. ABC + ADC = = 64 (M1) ADC x 2 so x = 128 (A1) reasons opposite angles of a cyclic quadrilateral add to 180, the angle at the centre of a circle is twice y=the angle at the circumference (C1) Question 2 (Total 3 marks) Diagram NOT accurately drawn B, C and D are points on the circumference of a circle, centre O. AB and AD are tangents to the circle. Angle DAB = 50

16 Work out the size of angle BCD. Give a reason for each stage in your working. ODA = 90 (angle between radius and tangent is 90 ) DAO = 25 DOA = 180 (90 25) = 65 (triangle adds to 180 ) DOB = 65 x 2 = 130 Reasons given (M1) (M1) (C1) BCD = 130/2 = 65 (angle at centre is double angle at circumference) (A1) (Total 4 marks) Question 3 B and C are points on the circumference of a circle, centre O. AB and AC are tangents to the circle. Angle BAC = 40. Find the size of angle BCO. ACO is the angle between tangent and radius is 90 (M1) OAC = 20 or AOC = 70 or BOC = 140 or ABC = ACB or BCA = Tangents from a point equal so BOA is congruent to AOC Base angles of an isosceles triangle BCO = 20 (A1) 2 (M1) = 70 (Total 3 marks) Total /10

17 Circle Theorems 2 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A and B are points on the circumference of a circle, centre O. AC and BC are tangents to the circle. Angle ACB = 36. Find the size of angle OBA. Give reasons for your answer. OBC = 90 or AOB = 144 or both angles CAB and CBA = 72 or ACO and BOC = 18 or AOC = 72 Tangent/radius is 90 BOA an isosceles triangle tangents from a point are equal for complete correct method one correct reason given answer 18 (M1) (C1) (A1) (Total 3 marks)

18 Question 2 * A, B, C and D are points on the circumference of a circle, centre O. Angle AOC = y. Find the size of angle ABC in terms of y. Give a reason for each stage of your working. ADC = (M1) (A1) 2 Two reasons given Angle at centre is twice the angle at the circumference (C1) Opposite angles in cyclic quadrilateral add to 180 (C1) OR Reflex AOC = y (M1) (A1) Two reasons given Angles round a point add to 360 (C1) Angle at centre is twice the angle at the circumference (C1) (Total 4 marks)

19 Question 3 B, C and D are points on the circumference of a circle, centre O. ABE and ADF are tangents to the circle. Angle DAB = 40 Angle CBE = 75 Work out the size of angle ODC. ABO = 90 or OBC = 15 or ADO = 90 or FDO = 90 or EBO = 90 (M1) Reflex angle BOD = 360 ( ) = 220 Or BCD = ( ) = 70 Or DBO = 90 (180-40)/2 = 20 Or BOC = 180 ( ) = 150 With valid reasons ODC = 55 (M1) (A1) (Total 3 marks) Total /10

20 Circle Theorems 3 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 ADB is the tangent at D to the circle, centre O. C is a point on the circumference. Angle OCD = 25. Calculate angle CDA. Show each step of your calculation. Not to scale [3]

21 Question 2 B, D and E are points on a circle, centre O. DE is a diameter. AC is the tangent to the circle at B. Angle BDE = 18. Not to scale

22 Question 3 A, B and C are points on a circle, centre O. TCD is a tangent to the circle. Angle BAC = 52. Not to scale Total / 10

23 Circle Theorems 4 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 (1) Opposite angles of a cyclic quadrilateral add up to 180 o (B1) (b) Work out the value of angle BAX. You must show all of your working. Angle CDB = = 37 (angles in a triangle add up to 180) Angle DBA = 37 (alternate angles) Angle CDA = = 105 (Opposite angles of a cyclic quadrilateral add up to 180 o ) Angle BDA = = 68 Angle BAX = 68 o (Alternate segment theorem) M1 M1 M1 M1 A1 (5)

24 Question 2 Write down a value for the following angles, and give a reason. Angle DBC = 37 o - angles in the same sector, subtended from the same point, are equal B1 B1 (2) Angle ADC = 90 as angle in a semi-circle. Angles in a triangle add up to 180, so Angle ACD = 53 o B1 B1 (2) Total / 10

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.

21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then