Circle theorems I — Paper 2 · IGCSE Mathematics - International 0607

12 A NOT TO 35° SCALE E B 15° D C A, B, C, D and E are points on the circle. Angle CAD = 35° and angle EBD = 15° . Find (a) angle CBD, Angle CBD = … [1] (b) angle CDE. Angle CDE = … [1]

2 marks

Mark scheme: 12(a) 35 1 12(b) 130 1 FT 165 – their (a)

This question in 0607/22 May/June 2017

9 D C p° 75° NOT TO SCALE B E q° A A, B, C, D and E lie on the circle. Angle BCE = 75˚. Find the value of p and the value of q. p = … q = … [2]

2 marks

Mark scheme: 9 [p =] 75 2 B1 for each [q=]105

This question in 0607/23 May/June 2017

9 NOT TO SCALE O 140° D A 25° B C A, B, C and D are points on a circle centre O. Find (a) angle ACD, Angle ACD = … [2] (b) angle BAD. Angle BAD = … [2]

4 marks

Mark scheme: 9(a) 110 2 B1 for reflex angle AOD = 220 or AXD = 70 9(b) 45 2 FT 155 – their 110 B1 for angle BCD = their 110 + 25

This question in 0607/21 Oct/Nov 2017

10 D 9x° A C NOT TO SCALE 130° E 6x° B ABCD is a cyclic quadrilateral. DC is extended to E. Angle BCE = 130°, angle ABC = 6x° and angle ADC = 9x°. Find the value of (a) angle BAD, Angle BAD = … [1] (b) angle ABC. Angle ABC = … [2]

3 marks

Mark scheme: 10(a) 130 1 10(b) 72 2 M1 for 6x + 9x = 180 oe implied by 12 seen

This question in 0607/22 Oct/Nov 2017

7 NOT TO SCALE O 140° x° y° O is the centre of the circle. Find the value of x and the value of y. x = … y = … [2]

2 marks

Mark scheme: 7 [x = ] 70 2 B1 for each [y = ] 110 If 0 scored SC1 for their x + their y =180

This question in 0607/21 May/June 2018

12 A B Q NOT TO O SCALE 40° 20° D C P A, B, C and D are points on the circle centre O. PQ is a tangent to the circle at C. Find these angles. (a) Angle DAC Angle DAC = … [2] (b) Angle ABC Angle ABC = … [1] (c) Angle ACQ Angle ACQ = … [2] Question 13, 14 and 15 are printed on the next page.

5 marks

Mark scheme: 12(a) 70 2 B1 for ∠DOC = 140 (12b) 120 1 12(c) 60 2 B1 for reflex∠ COD = 220 or ∠OCA = 30

This question in 0607/22 Oct/Nov 2018

11 (a) Q T, U and V lie on a circle, centre O. PQ is a tangent to the circle at T. NOT TO TU is a diameter. T SCALE O x° Find the value of x and the value of y. U 42° y° P V x = … y = … [2] (b) D C 44° q° ABCD is a cyclic quadrilateral. NOT TO SCALE Find the value of p and the value of q. 85° p° A B p = … q = … [2]

4 marks

Mark scheme: 11(a) [x =] 48 2 B1 for each [y =] 42 FT 90 − their x 11(b) [p =] 44 2 B1 for each [q =] 51

This question in 0607/23 Oct/Nov 2018

15 a° b° NOT TO SCALE 25° 80° The diagram shows a cyclic quadrilateral. Find the value of a and the value of b. a = … b = … [2]

2 marks

Mark scheme: 15 [a = ] 25 2 B1 for each [b = ] 100

This question in 0607/22 May/June 2019

16 D NOT TO SCALE C y° A x° B O The diagram shows a circle, centre O. AOB is a straight line. BCD is a tangent to the circle at C. Find y in terms of x. y = … [3] Question 17 is printed on the next page.

3 marks

Mark scheme: 16 [ y = ]135 + 0.5 x 3 M2 for 180 − y = 45 − 0.5 x or M1 for 90 − x as angle at centre

This question in 0607/23 May/June 2019

12 (a) C NOT TO SCALE D x° 100° B A The points A, B, C and D lie on the circle. Find the value of x. x = … [1] (b) A NOT TO SCALE y° O B 36° C The points A, B and C lie on the circle, centre O. Find the value of y. y = … [1]

2 marks

Mark scheme: 12(a) 80 1 12(b) 72 1

This question in 0607/21 Oct/Nov 2019

9 B C NOT TO SCALE x° 70° T A A, B and C are points on a circle. TA is a tangent to the circle at A. CA = CB and angle BAT = 70°. Work out the value of x. x = … [2]

2 marks

Mark scheme: 9 55 2 B1 for angle C = 70 1 or M1 for (180 − theirC ) 2

This question in 0607/21 May/June 2020

9 P B Q 56° A 98° NOT TO SCALE C 57° D A, B, C and D are points on the circle. PBQ is a straight line. (a) Find angle DCB, giving a reason for your answer. Angle DCB = … because … … [2] (b) Is PBQ a tangent to the circle? Give a reason for your answer. … because … … [1]

3 marks

Mark scheme: 9(a) 82 2 B1 for each Opposite angles of a cyclic quadrilateral [add up to 180] oe 9(b) No and any mention of Alternate 1 Segment Theorem oe

This question in 0607/23 May/June 2020

12 A NOT TO SCALE 30° D 110° B P C Q The points A, B, C and D lie on a circle. PCQ is a tangent to the circle at C. Angle ABC = 110° and angle BAC = 30° . Find (a) angle ADC, Angle ADC = … [1] (b) angle ACP, Angle ACP = … [1] (c) angle PCB. Angle PCB = … [1]

3 marks

Mark scheme: 12(a) 70 1 12(b) 70 1 12(c) 30 1 their(b) – 40

This question in 0607/22 Oct/Nov 2020

9 (a) A E 76° NOT TO SCALE D B C A, B, C, and D are points on a circle. CDE is a straight line. Find angle ABC . Angle ABC = … [1] (b) Q P 25° NOT TO SCALE O R 40° T S P, Q, R, S and T are points on the circle centre O. TOQ is a straight line. (i) Find angle STR. Angle STR = … [1] (ii) Find angle QOR. Angle QOR = … [1]

3 marks

Mark scheme: 9(a) 76 1 9(b)(i) 25 1 9(b)(ii) 80 1

This question in 0607/22 Feb/March 2021

15 NOT TO S R SCALE Q A 70°70° B T Points Q, R, S and T lie on the circle. AB is a tangent to the circle at T. Angle RTB = 70˚. Find angle RQT. Angle RQT = … [2]

2 marks

Mark scheme: 15 110 2 B1 for angle RST = 70 or angle RTA = 110

This question in 0607/23 May/June 2021

13 A 40° T x° 77° NOT TO SCALE B AB is a tangent to the circle at T. Find the value of x. x = … [2]

2 marks

Mark scheme: 13 63 2 M1 for other angle at tangent = 63˚ or other angle in triangle = 40˚

This question in 0607/22 Oct/Nov 2021

12 (a) B A 32° NOT TO SCALE C D A, B, C, and D are points on a circle. Angle DAC = 32° . BC = DC. Find angle BCD. Angle BCD = … [2] (b) B D NOT TO O SCALE C 42° A E A, B and C are points on the circle centre O. ECD is a tangent to the circle at C. Angle ACE = 42° . Find angle AOC. Angle AOC = … [2]

4 marks

Mark scheme: 12(a) 116 2 B1 for angle DBC = 32⁰ allow angle BDC = 32⁰ 12(b) 84 2 B1 for angle ABC = 42⁰ soi or for angle OCA = 48⁰ soi

This question in 0607/21 May/June 2022

13 B A NOT TO SCALE C Q 55° D P A, B, C and D are points on the circle. PQ is a tangent to the circle at D. Angle BDQ = 55° . Complete these statements giving a reason for each answer. (a) Angle BAD = … because … … [2] (b) Angle BCD = … because … … [2]

4 marks

Mark scheme: 13(a) 55 2 B1 for each Alternate Segment 13(b) 125 2 FT 180 – their 55 Opposite angles in cyclic quadrilateral B1 for each

This question in 0607/22 Oct/Nov 2022

11 C NOT TO SCALE B 55° X E 25° 40° D A Y A, B, C and D are four points on a circle. AC and BD meet at E. XAY is a tangent to the circle at A. Find (a) angle CDB, Angle CDB = … [1] (b) angle ACB, Angle ACB = … [1] (c) angle DCE, Angle DCE = … [1] (d) angle YAD. Angle YAD = … [1]

4 marks

Mark scheme: 11(a) 25 1 11(b) 40 1 11(c) 60 1 11(d) 60 1 FT their (c)

This question in 0607/23 Oct/Nov 2022

12 B NOT TO SCALE 35° C E 22° A D F A, B, C and D are points on the circle. EF is a tangent to the circle at A. Angle DBC = 35° and angle ACD = 22° . Find (a) angle ABD Angle ABD = … [1] (b) angle ADC Angle ADC = … [1] (c) angle CAF. Angle CAF = … [1]

3 marks

Mark scheme: 12(a) 22 1 12(b) 123 1 FT 145 – their(a) 12(c) 57 1 FT 35 + their(a)

This question in 0607/22 May/June 2023

10 B NOT TO SCALE A 51° O Q C 64° D P A, B, C and D are points on the circle centre O. PDQ is a tangent to the circle at D. Angle BAD = 51° and angle PDA = 64° . Find (a) angle BCD Angle BCD = … [1] (b) angle ABD Angle ABD = … [1] (c) the obtuse angle BOD. Angle B O D = … [1]

3 marks

Mark scheme: 10(a) 129 1 10(b) 64 1 10(c) 102 1

This question in 0607/23 Oct/Nov 2023

10 B NOT TO SCALE O C A 124° D A, B, C, and D lie on a circle, centre O. Find (a) angle ABC Angle ABC = … [1] (b) obtuse angle AOC Angle AOC = … [1] (c) angle OCA. Angle OCA = … [1]

3 marks

Mark scheme: 10(a) 56 1 10(b) 112 1 FT 2their(a) 10(c) 34 1 180 −their(b) FT 2

This question in 0607/21 Oct/Nov 2024

12 B C x° NOT TO A O SCALE 260° A, B and C lie on the circle centre O. (a) Work out the value of x. x = … [2] (b) Y is a point on the minor arc BYC. Work out angle BYC. Angle BYC = … [1]

3 marks

Mark scheme: 12(a) 50 2 B1 for 100˚ seen at O 12(b) 130 1 FT 180 – their (a)

This question in 0607/22 Oct/Nov 2024

13 A F 75° NOT TO SCALE O E B D 113° C A, B, C, D, E and F are points on a circle, centre O. (a) Find angle BFE. Angle BFE = … [1] (b) Find angle BED. Angle BED = … [1] (c) Find reflex angle BOE. Give geometrical reasons for your answer. Angle BOE = … Reasons … … … [3]

5 marks

Mark scheme: 13(a) 75 1 13(b) 67 1 13(c) 210 3 B2 for 210 or B1 for 150 Angle at the centre is twice the angle at the circumference B1 for complete reasons and angles at a point add to 360

This question in 0607/21 May/June 2025