Angles — Paper 2 · IGCSE Mathematics 0580

16 For Examiner's B Use North NOT TO 3 km North SCALE 3 km C 85° A A, B and C are three places in a desert. Tom leaves A at 06 40 and takes 30 minutes to walk directly to B, a distance of 3 kilometres. He then takes an hour to walk directly from B to C, also a distance of 3 kilometres. (a) At what time did Tom arrive at C? Answer (a) [1] (b) Calculate his average speed for the whole journey. Answer (b) km/h [2] (c) The bearing of C from A is 085°. Find the bearing of A from C. Answer (c) [1]

4 marks

Mark scheme: 16 (a) (0)810 or 8:10 etc. 1 (b) 4 2 M1 (3 + 3)/(1 + 0.5) (c) 265 1

This question in 0580/21 Oct/Nov 2009

19 For Examiner's Use NOT TO B D 68° SCALE O C E F A Points A, B and C lie on a circle, centre O, with diameter AB. BD, OCE and AF are parallel lines. Angle CBD = 68°. Calculate (a) angle BOC, Answer(a) Angle BOC = [2] (b) angle ACE. Answer(b) Angle ACE = [2]

4 marks

Mark scheme: 19 (a) 44 2 M1 OCB = 68 (b) 158 2

This question in 0580/21 Oct/Nov 2009

16 For Examiner's B Use North NOT TO 3 km North SCALE 3 km C 85° A A, B and C are three places in a desert. Tom leaves A at 06 40 and takes 30 minutes to walk directly to B, a distance of 3 kilometres. He then takes an hour to walk directly from B to C, also a distance of 3 kilometres. (a) At what time did Tom arrive at C? Answer (a) [1] (b) Calculate his average speed for the whole journey. Answer (b) km/h [2] (c) The bearing of C from A is 085°. Find the bearing of A from C. Answer (c) [1]

4 marks

Mark scheme: 16 (a) (0)810 or 8:10 etc. 1 (b) 4 2 M1 (3 + 3)/(1 + 0.5) (c) 265 1

This question in 0580/22 Oct/Nov 2009

19 For Examiner's Use NOT TO B D 68° SCALE O C E F A Points A, B and C lie on a circle, centre O, with diameter AB. BD, OCE and AF are parallel lines. Angle CBD = 68°. Calculate (a) angle BOC, Answer(a) Angle BOC = [2] (b) angle ACE. Answer(b) Angle ACE = [2]

4 marks

Mark scheme: 19 (a) 44 2 M1 OCB = 68 (b) 158 2

This question in 0580/22 Oct/Nov 2009

17 A NOT TO E SCALE O D 38° C B AB is the diameter of a circle, centre O. C, D and E lie on the circle. EC is parallel to AB and perpendicular to OD. Angle DOC is 38°. Work out (a) angle BOC , Answer(a) Angle BOC = [1] (b) angle CBO, Answer(b) Angle CBO = [1] (c) angle EDO . Answer(c) Angle EDO = [2]

4 marks

Mark scheme: 17 (a) 52 1 (b) 64 1 (c) 71 2 M1 angle CED = 19

This question in 0580/21 May/June 2010

4 D For Examiner's NOT TO Use C SCALE 50° A B O O is the centre of the circle. DA is the tangent to the circle at A and DB is the tangent to the circle at C. AOB is a straight line. Angle COB = 50°. Calculate (a) angle CBO, Answer(a) Angle CBO = [1] (b) angle DOC. Answer(b) Angle DOC = [1]

2 marks

Mark scheme: 4 (a) 40 1 (b) 65 1 20 50

This question in 0580/22 Oct/Nov 2010

10 For A B Examiner's Use NOT TO 140° 140° SCALE E C 140° D The pentagon has three angles which are each 140°. The other two interior angles are equal. Calculate the size of one of these angles. Answer [3]

3 marks

Mark scheme: 10 60 3 B1 540 used M1 [their 540 – 3 × 140]/2 2

This question in 0580/22 Oct/Nov 2010

9 A B 2x° NOT TO SCALE C D x° 5x° AB is parallel to CD. Calculate the value of x. Answer x = [3]

3 marks

Mark scheme: 9 22.5 oe 3 B2 180 = 5x +2x + x oe or better B1 for 2x or 6x marked in the correct place on the diagram.

This question in 0580/21 May/June 2011

20 For V Examiner's Use U NOT TO 70° SCALE W g° O h° X e° f ° A T B The diagram shows a circle, centre O. VT is a diameter and ATB is a tangent to the circle at T. U, V, W and X lie on the circle and angle VOU = 70°. Calculate the value of (a) e, Answer(a) e = [1] (b) f, Answer(b) f = [1] (c) g, Answer(c) g = [1] (d) h. Answer(d) h = [1]

4 marks

Mark scheme: 20 (a) 35 1 (b) 55 1ft 90 – (a) but b > 0 (c) 55 1ft = (b) (d) 125 1ft 180 – (c)

This question in 0580/23 May/June 2011

13 For North Examiner's T Use B NOT TO SCALE 76° A C O P AOC is a diameter of the circle, centre O. AT is a straight line that cuts the circle at B. PT is the tangent to the circle at C. Angle COB = 76°. (a) Calculate angle ATC. Answer(a) Angle ATC = [2] (b) T is due north of C. Calculate the bearing of B from C. Answer(b) [2]

4 marks

Mark scheme: 13 (a) 52 2 M1 OAB or OBA = 38 or OCT = 90 (b) 322 2 M1 BCT = 38 or BCO = 52 IGCSE – October/November 2011 0580 22

This question in 0580/22 Oct/Nov 2011

9 D A B (a) The point C lies on AD and angle ABC = 67°. Draw accurately the line BC. [1] (b) Using a straight edge and compasses only, construct the perpendicular bisector of AB. Show clearly all your construction arcs. [2]

3 marks

Mark scheme: 9 (a) 1 B1 C marked on AD unless the line stops at AD and angle of 67° at B also correct ruled line (b) 2 B1 correct arcs B1 correct ruled line perpendicular bisector of AB 750 × 5 × 2 5

This question in 0580/21 May/June 2012

1 For A Examiner's Use 73° B NOT TO 120° SCALE 82° x° C D E The diagram shows a quadrilateral ABCD. CDE is a straight line. Calculate the value of x. Answer x = [2]

2 marks

Mark scheme: Qu Answers Mark Part marks 1 95 2 B1 for 85 seen or M1 x = 180 – their angle ADC, if it is clearly seen 750 × 2 × 8 M1 for oe seen or SC1 870 as final

This question in 0580/23 May/June 2012

4 50° NOT TO 55° SCALE a° Use the information in the diagram to fi nd the value of a. Answer a = … [2] _____________________________________________________________________________________

2 marks

Mark scheme: 4 105 2 M1 for 180 – 55 – 50 or B1 for 55 or 75 seen in the correct angle inside the triangle 3 k

This question in 0580/21 May/June 2013

3 NOT TO SCALE 55° p° Find the value of p. Answer p = … [2] _____________________________________________________________________________________

2 marks

Mark scheme: 3 125 2 B1 for 55 or 125 in any other correct position on diagram or M1 for 180–55

This question in 0580/21 Oct/Nov 2013

13 C NOT TO SCALE O B 58° D 23° A A, B, C and D lie on a circle centre O. Angle ABC = 58° and angle CAD = 23°. Calculate (a) angle OCA, Answer(a) Angle OCA = … [2] (b) angle DCA. Answer(b) Angle DCA = … [2] __________________________________________________________________________________________

4 marks

Mark scheme: 13 (a) 32 2 B1 for AOC = 116 (b) 35 2 B1 for CDA = 122

This question in 0580/21 May/June 2014

19 C NOT TO SCALE A B D Two circles, centres A and B, are each of radius 8 cm and intersect at C and D. Each circle passes through the centre of the other circle. (a) Explain why angle CBD is 120°. Answer(a) [1] C (b) For the circle, centre B, fi nd the area of the sector BCD. NOT TO 8 cm SCALE A 120° B 8 cm D Answer(b) … cm2 [2] (c) (i) Find the area of the shaded segment CAD. C NOT TO SCALE A B D Answer(c)(i) … cm2 [3] (ii) Find the area of overlap of the two circles. Answer(c)(ii) … cm2 [1] __________________________________________________________________________________________ Question 20 is printed on the next page.

7 marks

Mark scheme: 19 (a) CBA and BDA are equilateral oe 1 (b) 67[.0] or 67.02 to 67.03 2 M1 for 120360 × π × 8 2 oe (c) (i) 3 M2FT for their (b ) − 12 × 82 × sin 120 oe 39.3 or 39.28 to 39.33 or M1 for 12 × 82 × sin 120 oe (ii) 1FT FT 2 × their(c)(i) correctly evaluated 78.6 or 78.7 or 78.56 to 78.66

This question in 0580/21 Oct/Nov 2014

20 (a) W V NOT TO 88° SCALE X Z 57° Y Two straight lines VZ and YW intersect at X. VW is parallel to YZ, angle XYZ = 57° and angle VXW = 88°. Find angle WVX. Answer(a) Angle WVX = … [2] (b) A NOT TO 7.2 cm SCALE P Q 8.4 cm B C 12.6 cm ABC is a triangle and PQ is parallel to BC. BC = 12.6 cm, PQ = 8.4 cm and AQ = 7.2 cm. Find AC. Answer(b) AC = … cm [2] __________________________________________________________________________________________

4 marks

Mark scheme: 20 (a) 35 2 M1 for [Z =] 180 − 88 − 57 or VWX = 57 or YZX = 35 AC 126. (b) 10.8 2 M1 for = oe 2.7 4.8

This question in 0580/22 Feb/March 2015

8 P 11 cm NOT TO SCALE 37° A O In the diagram, AP is a tangent to the circle at P. O is the centre of the circle, angle PAO = 37° and AP = 11 cm. (a) Write down the size of angle OPA. Answer(a) Angle OPA = … [1] (b) Work out the radius of the circle. Answer(b) … cm [2] __________________________________________________________________________________________

3 marks

Mark scheme: 8 (a) 90 1 OP o (b) 8.29 or 8.289… to 8.29 2 M1 for = tan 37 oe 11

This question in 0580/21 Oct/Nov 2015

8 Find the sum of the interior angles of a 25-sided polygon. Answer … [2] __________________________________________________________________________________________

2 marks

Mark scheme:  360 8 4140 2 M1 for ( 25 − 2) × 180 or 25 ×  180 −   25  2

This question in 0580/23 Oct/Nov 2015

18 (a) x° NOT TO SCALE 47° Find the value of x. x = … [1] (b) 85° 115° NOT TO SCALE 97° y° Find the value of y. y = … [2] (c) 58° NOT TO z° SCALE O The diagram shows a circle, centre O. Find the value of z. z = … [2]

5 marks

Mark scheme: 18 (a) 47 1 (b) 117 2 M1 for 360 − (115 + 85 + 97) (c) 244 2 B1 for 116 seen at centre or 122 seen at circumference

This question in 0580/22 Feb/March 2016

9 C D 40° NOT TO SCALE b° a° A B Triangle ABC is isosceles and AC is parallel to BD. Find the value of a and the value of b. a = … b = … [2]

2 marks

Mark scheme: 9 [a = ] 70 2 B1 for each [b = ] 40

This question in 0580/21 May/June 2016

17 Five angles of a hexagon are each 115°. Calculate the size of the sixth angle. … [3]

3 marks

Mark scheme: 17 145 3 M2 for (6 – 2) × 180 – 5 × 115 or M1 for (6 – 2) × 180 Alt method M2 for 180 – (360 – 5 × (180 – 115)) or M1 for 360 – 5 × (180 – 115) 1000

This question in 0580/21 May/June 2016

12 B 45° NOT TO SCALE C 20° A E D ABCE is a cyclic quadrilateral. AED and BCD are straight lines. AC = CD, angle ABC = 45° and angle ACE = 20°. Work out angle ECD. Angle ECD = … [3]

3 marks

Mark scheme: 12 110 3 B2 for ADC = 25 or B1 for AEC = 135 or CAE = 25

This question in 0580/23 May/June 2016

13 NOT TO SCALE x° y° 42° The diagram is made from 5 congruent kites. Work out the value of (a) x, x = … [1] (b) y. y = … [2]

3 marks

Mark scheme: 13 (a) 72 1 (b) 123 2FT FT dep. on answer being obtuse M1 for (360 – their ( a ) – 42) [÷2]

This question in 0580/23 May/June 2016

2 67° NOT TO SCALE a° 42° Find the value of a. a = … [2]

2 marks

Mark scheme: 2 25 2 B1 for 67 or 113 seen once in correct position or M1 for a + 42 = 67 or a + 42 + 113 = 180 or better

This question in 0580/22 Oct/Nov 2016

15 (a) A NOT TO SCALE C 44° B Triangle ABC is an isosceles triangle with AB = CB. Angle ABC = 44°. Find angle ACB. Angle ACB = … [1] (b) A regular polygon has an exterior angle of 40°. Work out the number of sides of this polygon. … [2]

3 marks

Mark scheme: 15 (a) 68 1 (b) 9 2 M1 for 360 ÷ 40 oe or 180 ( n − 2 ) = 140 oe n k

This question in 0580/23 Oct/Nov 2016

8 NOT TO p° 70° SCALE q° The diagram shows a straight line intersecting two parallel lines. Find the value of p and the value of q. p = … q = … [2]

2 marks

Mark scheme: 8 110 1 70 1

This question in 0580/22 May/June 2017

10 The three angles in a triangle are 5x°, 6x° and 7x°. 7x° NOT TO SCALE 6x° 5x° (a) Find the value of x. x = … [2] (b) Work out the size of the largest angle in the triangle. … [1]

3 marks

Mark scheme: 10(a) 10 2 180 M1 for 5 x + 6 x + 7 x = 180 oe or 5 + 6 + 7 or B1 for angles 50, 60 and 70 10(b) 70 1FT FT 7 × their (a) provided 0 < their answer < 180

This question in 0580/23 May/June 2017

1 NOT TO 72° 83° SCALE 104° x° The diagram shows a quadrilateral. Find the value of x. x = … [1]

1 marks

Mark scheme: Question Answer Mark Partial marks 1 101 1

This question in 0580/21 Oct/Nov 2017

11 The diagram shows a regular pentagon. A AB is a line of symmetry. d ° Work out the value of d. NOT TO SCALE B d = … [3]

3 marks

Mark scheme: 11 54 3 180 × ( 5 − 2 ) 360 M2 for or 180 − 5 5 360 or M1 for 180 × (5 − 2) or 5

This question in 0580/21 Oct/Nov 2017

5 NOT TO 80° SCALE 120° y° x° A B In the diagram, AB is a straight line. Find the value of x and the value of y. x = … y = … [2]

2 marks

Mark scheme: 5 [x = ] 60 2 B1 for each or for two numbers that add to 100 [y =] 40

This question in 0580/22 Oct/Nov 2017

17 NOT TO 29x° SCALE x° The diagram shows part of a regular polygon. The exterior angle is x°. The interior angle is 29x°. Work out the number of sides of this polygon. … [3]

3 marks

Mark scheme: 17 60 3 B2 for x = 6 or M1 for 29x + x = 180 oe and M1 for 360 ÷ 6 or 360 ÷ their x or 180(n – 2) = their x × 29n

This question in 0580/22 Oct/Nov 2017

5 Q T A 43° C x° NOT TO SCALE P B S The diagram shows two parallel lines PAQ and SBCT. AB = AC and angle QAC = 43°. Find the value of x. x = … [2]

2 marks

Mark scheme: 5 94 2 B1 for ACB or PAB or ABC = 43 or M1 for 180 −×2 43 or 12 x = 90 − 43

This question in 0580/21 May/June 2018

7 A and B are two towns on a map. The bearing of A from B is 140°. Work out the bearing of B from A. … [2]

2 marks

Mark scheme: 7 320 2 M1 for 180 + 140 oe

This question in 0580/22 May/June 2018

16 A B NOT TO x° SCALE 56° L M C y° N The diagram shows an isosceles triangle ABC with AB = AC. LCM and BCN are straight lines and LCM is parallel to AB. Angle ACL = 56°. Find the value of x and the value of y. x = … y = … [4]

4 marks

Mark scheme: 16 [x =] 62 2 B1 for 56 identified as angle A (180 − 56 ) or M1 for 2 [y =] 118 2 FT for 2 marks their acute x + their y =180 or 56 + their acute x = their y or B1 for any of ACB, BCM or LCN = 62 or their acute x or M1 for 180 – 62 or 180 – their acute x or 56 + 62 or 56 + their acute x

This question in 0580/21 Oct/Nov 2018

4 Complete each statement. (a) A quadrilateral with only one pair of parallel sides is called a … . [1] (b) An angle greater than 90° but less than 180° is called … . [1]

2 marks

Mark scheme: 4(a) Trapezium 1 4(b) Obtuse 1

This question in 0580/22 May/June 2019

25 K J NOT TO E D SCALE O A C F B H G The diagram shows two regular pentagons. Pentagon FGHJK is an enlargement of pentagon ABCDE, centre O. (a) Find angle AEK. Angle AEK = … [4] (b) The area of pentagon FGHJK is 73.5 cm2. The area of pentagon ABCDE is 6 cm2. Find the ratio perimeter of pentagon FGHJK : perimeter of pentagon ABCDE in its simplest form. … : … [2]

6 marks

Mark scheme: 25(a) 126 4 360 − 180 – ( 360 ÷ 5 )  M3 for or 2 360 − 180 × ( 5 − 2 ) ÷ 5 2 180 × ( 5 − 2 ) 360 or M2 for or 180 – 5 5 360 or M1 for 180 × (5 – 2) or 5 25(b) 7 : 2 2 73.5 6 M1 for or 6 73.5

This question in 0580/23 May/June 2019

4 Find the interior angle of a regular polygon with 24 sides. … [2]

2 marks

Mark scheme: 4 165 2 ( 24 − 2 ) × 180 360 M1 for or 180 − 24 24

This question in 0580/22 Feb/March 2020

6 The diagram shows a trapezium. ( 97 - 3x)° NOT TO SCALE ( 69 + 5x)° Work out the value of x. x = … [3]

3 marks

Mark scheme: 6 7 3 M2 for 166 + 2x = 180 or better or M1 for 97 – 3x + 69 + 5x = 180 oe

This question in 0580/21 May/June 2020

3 C NOT TO x° SCALE 50° A B D AB = BC and ABD is a straight line. Find the value of x. x = … [2]

2 marks

Mark scheme: 3 25 2 B1 for 130 seen or M1 for 50 ÷ 2

This question in 0580/22 May/June 2020

9 North NOT TO SCALE North A B The bearing of B from A is 105°. Find the bearing of A from B. … [2]

2 marks

Mark scheme: 9 285 2 M1 for 180 + 105 or 75 or 105 seen in correct position at B

This question in 0580/22 May/June 2020

3 A NOT TO SCALE 32° B C Triangle ABC is isosceles. Angle ABC = 32° and AB = AC. Find angle BAC. Angle BAC = … [2]

2 marks

Mark scheme: 3 116 2 M1 for angle ACB = 32 soi

This question in 0580/23 May/June 2020

12 The interior angle of a regular polygon with n sides is 156°. Work out the value of n. n = … [2]

2 marks

Mark scheme: 12 15 2 360 180 ( n − 2 ) M1 for or =156 oe 180 − 156 n

This question in 0580/21 Oct/Nov 2020

4 x° y° NOT TO SCALE 140° 120° The diagram shows a triangle drawn between a pair of parallel lines. Find the value of x and the value of y. x = … y = … [3]

3 marks

Mark scheme: 4 [x =] 60 3 B1 for [x =] 60 [y =] 80 B2 for [y =] 80 or B1 for 40 in a correct place on diagram If 0 scored SC1 for their x + their y = 140

This question in 0580/22 Oct/Nov 2020

19 100° NOT TO 8 cm SCALE x° 9 cm (a) Calculate the value of x. x = … [3] (b) Calculate the area of the triangle. … cm2 [3]

6 marks

Mark scheme: 19(a) 61.1 or 61.08 to 61.09... 3 8sin100 M2 for [sin x =] oe or better 9 9 8 or M1 for = oe sin100 sin x 19(b) 11.7 or 11.66 to 11.67 3 M2 for 1 × 9 × 8 × sin(180 − 100 − their (a)) oe 2 or M1 for 180 – 100 – their (a)

This question in 0580/22 Oct/Nov 2020

6 A NOT TO 58° SCALE 82° B C The diagram shows triangle ABC. The triangle is reflected in the line BC to give a quadrilateral ABDC. (a) Write down the mathematical name of the quadrilateral ABDC. … [1] (b) Find angle ACD. Angle ACD = … [2]

3 marks

Mark scheme: 6(a) Kite 1 6(b) 80 2 M1 for (180 – 82 – 58) or better

This question in 0580/23 Oct/Nov 2020

4 NOT TO 59° 37° SCALE a° c° b° The diagram shows two parallel lines intersected by two straight lines. Find the values of a, b and c. a = … b = … c = … [3]

3 marks

Mark scheme: 4 [a =] 59 3 B1 for each [b =] 37 [c =] 84 If 0 scored SC1 for their (a + b + c) = 180 if a, b, c > 0

This question in 0580/21 May/June 2021

3 x NOT TO SCALE 40° The diagram shows a pair of parallel lines and a straight line. Complete the statement with the correct geometrical reason. x = 40° because the angles are … [1]

1 marks

Mark scheme: 3 Corresponding 1

This question in 0580/23 May/June 2021

4 NOT TO SCALE 100° y° y° Find the value of y. y = … [2]

2 marks

Mark scheme: 4 130 2 M1 for 360 – 100 or better

This question in 0580/23 May/June 2021

6 The scale drawing shows the positions of two towns, P and Q. The scale is 1 cm represents 4 km. North Q North P Scale: 1 cm to 4 km (a) Find the actual distance between town P and town Q. … km [2] (b) Measure the bearing of town Q from town P. … [1] (c) Town X is 28 km from town P on a bearing of 140°. On the scale drawing, mark the position of town X. [2]

5 marks

Mark scheme: 6(a) 32.8 2 M1 for 8[cm] to 8.4[cm] seen or for their measurement [in cm] multiplied by 4 6(b) 065 1 6(c) X correctly placed 7 cm from P 2 M1 for X on bearing of 140 from P on a bearing of 140° or for X 7 cm from P If 0 scored SC1 for X on bearing of 140 from Q and 7 cm from Q

This question in 0580/21 Oct/Nov 2021

4 A 86° B x° NOT TO SCALE C 58° D Triangle ABC and triangle ACD are isosceles. Angle DAB = 86° and angle ADC = 58°. Find the value of x. x = … [3]

3 marks

Mark scheme: 4 79 nfww 3 M2 for x + x + 58 + 58 + 86 = 360 oe or 86 – (180 – 2 × 58) implied by CAB = 22 or B1 for DCA = 58 or BCA = x or DAC = 64

This question in 0580/22 Oct/Nov 2021

2 x° NOT TO SCALE 132° The diagram shows two parallel lines intersecting a straight line. Find the value of x. x = … [2]

2 marks

Mark scheme: 2 48 2 B1 for 132 or 48 in the correct position on the diagram or M1 for 180 – 132

This question in 0580/23 Oct/Nov 2021

1 Measure the marked angle. … [1]

1 marks

Mark scheme: Question Answer Marks Partial Marks 1 40° 1

This question in 0580/22 Feb/March 2022

9 North B NOT TO SCALE A The bearing of B from A is 059°. Work out the bearing of A from B. … [2]

2 marks

Mark scheme: 9 239 2 M1 for 180 + 59 or 360 – (180 − 59) oe or indicates correct angle on diagram

This question in 0580/22 May/June 2022

3 T NOT TO S SCALE R 73° 75° P 129° Q PQRS is a quadrilateral. RST is a straight line. Find angle PST. Angle PST = … [2]

2 marks

Mark scheme: 3 97 2 M1 for 360 – (73 + 129 + 75)

This question in 0580/23 May/June 2022

5 x° 71° NOT TO SCALE 55° The diagram shows two straight lines intersecting two parallel lines. Find the value of x. x = … [2]

2 marks

Mark scheme: 5 54 2 M1 for 180 – 71 – 55 oe or B1 for 55 or 125 in a relevant correct position on the diagram

This question in 0580/21 Oct/Nov 2022

1 x° NOT TO SCALE 34° The diagram shows an isosceles triangle. Find the value of x. x = … [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 112 2 M1 for 180 – 34  2 oe

This question in 0580/22 Oct/Nov 2022

11 NOT TO SCALE (7x + 44)° (x + 8)° The diagram shows two sides of a regular polygon. The interior angle of the polygon is ( 7x + 44)° and the exterior angle is ( x + 8 )° . Find the number of sides of this polygon. … [4]

4 marks

Mark scheme: 11 15 4 B2 for x = 16 soi or M1 for 7x + 44 + x + 8 = 180 or better M1 for 360 ÷ (their x + 8) oe

This question in 0580/23 Oct/Nov 2022

1 B A 104° 71° NOT TO SCALE 56° C E D CDE is a straight line. Find angle ADE. … [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 51 2 M1 for 360 – (56 + 104 + 71)

This question in 0580/21 May/June 2023

3 58° a° NOT TO SCALE b° The diagram shows a straight line intersecting two parallel lines. Find the value of a and the value of b, giving a geometrical reason for each answer. a = … because … b = … because … [4]

4 marks

Mark scheme: 3 58, vertically opposite 2 B1 for each 122, interior 2 B1 for each

This question in 0580/21 May/June 2023

1 The diagram shows an isosceles triangle. 41° NOT TO SCALE x° Find the value of x. x = … [2]

2 marks

Mark scheme: Question Answer Marks Partial Marks 1 98 2 M1 for x + 41 + 41 = 180 oe or better

This question in 0580/21 Oct/Nov 2023

5 50° x° NOT TO SCALE 114° The diagram shows two intersecting straight lines crossing two parallel lines. Find the value of x. x = … [2]

2 marks

Mark scheme: 5 64 2 B1 for any of these angles labelled on the diagram or M1 for x + 50 = 114 or better

This question in 0580/22 Oct/Nov 2023

5 B NOT TO 112° SCALE 44° C M A The diagram shows triangle ABC. M is the midpoint of AC. Triangle ABC is rotated 180° about centre M. The image and the original triangle together form a quadrilateral ABCD. (a) Write down the mathematical name of the quadrilateral ABCD. … [1] (b) Find angle BAD. Angle BAD = … [2]

3 marks

Mark scheme: 5(a) Parallelogram 1 5(b) 68 2 M1 for 180 – 112 oe or for 180 − 112 − 44

This question in 0580/23 Oct/Nov 2023

6 D A 40° C E B NOT TO SCALE x° The diagram shows 5 kites that are congruent to kite ABCD. Each kite is joined to the next kite along one edge. Angle DAB = 40° and DCE is a straight line. Find the value of x. x = … [3]

3 marks

Mark scheme: 6 145 3 M1 for 180 ÷ 6 or any angle congruent to BCD = 30 M1 for (360 – 40 – their 30) ÷ 2 oe

This question in 0580/21 May/June 2024

15 North A NOT TO C SCALE B Three towns, A, B and C, are equidistant from each other. The bearing of C from A is 104°. Calculate the bearing of B from C. … [3]

3 marks

Mark scheme: 15 224 3 M2 for a fully correct method e.g. 360 – (180 – 104 + 60) oe or B2 for 120, 136, 44, 46, 14, or 16 in the correct position or B1 for 60, 76, 104 or 284 in the correct position or for interior angle of triangle = 60 i.e. these positions for B2 or B1: 44 76 60 60 60 136 76 44 16 60 44 104 104 120 14 46 284

This question in 0580/22 May/June 2024

18 B NOT TO SCALE A X V The diagram shows two sides, VA and VB, of a regular polygon. AVX is a straight line. Angle BVX = y° and angle AVB = 11.5 y° . Find the number of sides of this polygon. … [3]

3 marks

Mark scheme: 18 25 3 B2 for [y =] 14.4 oe or M1 for y + 11.5y = 180 or for 360 ÷ their y

This question in 0580/23 May/June 2024

6 The diagram shows a right-angled triangle ABC and a quadrilateral AEDC. C y° 17° NOT TO SCALE D z° 122° x° 34° A B E Find the value of (a) x x = … [1] (b) y y = … [1] (c) z. z = … [1]

3 marks

Mark scheme: 6(a) 58 1 6(b) 39 1 6(c) 251 1

This question in 0580/21 Oct/Nov 2024

9 NOT TO SCALE `x + 132j ° x° The diagram shows part of a regular polygon. The interior angle of the polygon is 132° larger than the exterior angle. Calculate the number of sides of this polygon. … [3]

3 marks

Mark scheme: 9 15 3 B2 for [x =] 24 OR M1 for x + x + 132 = 180 oe soi 360 M1 for oe provided this gives an their x integer answer

This question in 0580/21 Oct/Nov 2024

2 The diagram shows an isosceles triangle. x° NOT TO SCALE 43° Find the value of x. x = … [2]

2 marks

Mark scheme: 2 94 2 M1 for x + 2 × 43 = 180 oe

This question in 0580/22 Oct/Nov 2024

8 The diagram shows a parallelogram. (132 – 2x)° NOT TO SCALE (15 + 5x)° Work out the size of the smallest interior angle of the parallelogram. … [4]

4 marks

Mark scheme: 8 70 4 B3 for x = 11 OR M1 for 132 – 2x + 15 + 5x = 180 oe M1 for collecting x terms on one side and number terms on the other for their equation. M1 for 15 + 5 × their x oe where –3 < their x < 15 or for 132 – 2 × their x oe where 21 < their x < 66

This question in 0580/22 Oct/Nov 2024

2 x° NOT TO SCALE w° y° 158° 76° The diagram shows two parallel lines intersecting two straight lines. Find the values of w, x and y. w = … x = … y = … [4]

4 marks

Mark scheme: 2 [w =] 158 4 B1 for w correct [x =] 76 B1 for x correct [y =] 82 B2FT for y = 158 – their x correctly evaluated or B1 for 22 (identified) or 82 in position vertically opposite to y or for y = 158 – their x

This question in 0580/21 May/June 2025

2 The scale drawing shows the positions of two villages, P and Q. The scale is 1 cm represents 0.5 km. North P North Q (a) Find the actual distance between village P and village Q. … km [2] (b) Measure the bearing of village Q from village P. … [1]

3 marks

Mark scheme: 2(a) 4.4 to 4.6 2 B1 for 8.8 [cm] to 9.2 [cm] or M1 for 0.5 × their written measurement where their measurement is in the range 8 to 10 2(b) 108 to 112 1

This question in 0580/22 May/June 2025

3 45° 65° NOT TO SCALE x° y° The diagram shows two straight lines intersecting two parallel lines. Find the value of x and the value of y. x = … y = … [3]

3 marks

Mark scheme: 3 [x =] 70 3 B2 for either correct and or B1 for 45 and 65 correctly placed on [y =] 65 diagram or M1 for 180 – 45 – 65 oe

This question in 0580/22 May/June 2025

12 NOT TO R SCALE P 74° Q P, Q and R lie on a circle. QR is a diameter. Find angle PRQ. Give geometrical reasons for your answer. Angle PRQ = … because … … [2]

2 marks

Mark scheme: 12 16 2 B1 for 16 or angle in a semicircle = 90 and angle in a semicircle = 90 and angle sum of a triangle = 180

This question in 0580/22 May/June 2025

2 A x° E 30° y° F 104° NOT TO SCALE D 130° B 70° C ABCD is a quadrilateral. CDE is a straight line. AFB is an isosceles triangle. Find the value of x and the value of y. x = … y = … [4]

4 marks

Mark scheme: 2 x = 84 4 B2 for x = 84 y = 75 or M1 for CDA = 76 or 360 – 70 – 130 – (180 – 104) oe B2 for y = 75 or M1 for (180 – 30) ÷ 2

This question in 0580/23 May/June 2025

2 148° NOT TO 82° B SCALE C A In the diagram, AC = BC. Work out angle CAB. Angle CAB = … [3]

3 marks

Mark scheme: 2 25 3 M1 for 360 – 148 – 82 M1 for (180 – their 130) ÷ 2

This question in 0580/21 Oct/Nov 2025

3 Find the interior angle of a regular 20-sided polygon. … [2]

2 marks

Mark scheme: 3 162 2 360 180 ( 20 − 2 ) M1 for 180 – or for 20 20

This question in 0580/21 Oct/Nov 2025

14 C 55° NOT TO B SCALE X 88° A D E A, B, C, D and E lie on the circle. AC and BD intersect at X. Angle ACD = 55° and angle CXD = 88°. (a) Complete the statements, giving a geometrical reason in each part. Angle CDB = … because … … Angle ABD = … because … … Angle AED = … because … … [6] (b) Triangle CXD is mathematically similar to triangle BXA. DX = 8.0 cm, BX = 2.7 cm and AX = 4.0 cm. (i) Work out the length of CX. CX = … cm [2] (ii) Complete the statement. Area of triangle CXD : area of triangle BXA = … : … [1]

9 marks

Mark scheme: 14(a) 37 2 B1 for each Angle sum of triangle = 180 55 2 B1 for each Angles in the same segment are equal 125 2 B1 for each Opposite angles of cyclic quadrilateral sum to 180 14(b)(i) 5.4 2 4 2.7 M1 for = oe 8 CX 14(b)(ii) 4 : 1 oe 1

This question in 0580/21 Oct/Nov 2025

6 B A NOT TO 75° SCALE 80° C E D ABCD is a quadrilateral. E lies on CD and AE is parallel to BC. EA = ED. Find (a) angle ABC Angle ABC = … [1] (b) angle AED Angle AED = … [1] (c) angle DAB. Angle DAB = … [2]

4 marks

Mark scheme: 6(a) 105 1 6(b) 80 1 6(c) 125 2 FT (180 – their (b)) ÷ 2 + 75 B1 for EAD or EDA = 50 or B1FT for 180 − their (b) EAD or EDA = correctly 2 evaluated or M1 for (180 – their (b)) ÷ 2 + 75

This question in 0580/22 Oct/Nov 2025

16 D NOT TO C SCALE 14° A B AB and BD are two sides of a regular 15-sided polygon. AB and BC are two sides of a regular n-sided polygon. Angle DBC = 14°. Work out the value of n. n = … [4]

4 marks

Mark scheme: 16 36 4  360  M3 for 360   − 14  oe  15  or B2 for 24 or 156 (15 − 2 ) 180 or M1 for 360 ÷ 15 or oe 15

This question in 0580/22 Oct/Nov 2025

5 B F C 12º xº E NOT TO SCALE yº A G D The diagram shows a rectangle ABCD. EFG is an equilateral triangle that touches the rectangle at F and G. Find the value of x and the value of y. x = … y = … [2]

2 marks

Mark scheme: 5 x = 108 2 B1 for x = 108 or y = 72 y = 72 or their x + their y = 180

This question in 0580/23 Oct/Nov 2025

13 D NOT TO SCALE A 44° O C x° E B A, B, C and D are points on the circumference of a circle with centre O. ED and EB are tangents to the circle. AC is parallel to EB. Angle AOD = 44°. Find the value of x. x = … [4]

4 marks

Mark scheme: 13 23 4 M3 for x + x + 44 + 90 = 180 oe OR B1 for ODE or OBE or BOA = 90° B1 for AOE = x or OED = x

This question in 0580/23 Oct/Nov 2025