PAST PAPER
QUESTIONS
IGCSE/O LEVEL
0580/4024
MATHS
TOPIC: ANGLE
PROPERTIES OF CIRCLES
@MathleteBySaad
saad@mathletebysaad.com
Difficulty level: Easy
18
C
A
O
38°
B
T
A, B and C are points on the circumference of a circle centre O.
t = 38°.
O is the midpoint of BC and ABC
Tangents are drawn from T to touch the circle at A and B.
t .
(a) Calculate BCA
t = ............................. [1]
Answer BCA
t .
(b) Calculate ATB
t = .............................. [2]
Answer ATB
9
Find the integers that satisfy 1 1 3x + 5 G 11.
Answer .......................................... [2]
© UCLES 2018
4024/11/M/J/18
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'
17
222
A
53°
62°
O
x°
t°
D
y°
C
T
z°
B
The diagram shows a circle, centre O, that passes through A, B, C and D.
The tangents at A and B meet at T.
t = 53° .
t = 62° and DAB
ATB
(a) Find x.
Answer x = ..................................... [1]
(b) Find y.
Answer y = ..................................... [1]
(c) Find z.
Answer z = ..................................... [1]
(d) Find t.
Answer t = .................................... [1]
© UCLES 2018
4024/12/M/J/18
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'
7
10
3
E
NOT TO
SCALE
D
24°
F
y°
O
110°
x°
A
C
B
Points A, B, C, D, E and F lie on the circle, centre O.
Find the value of x and the value of y.
x = ..................................................
y = .................................................. [2]
© UCLES 2020
0580/21/M/J/20
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12
174
A
B
NOT TO
SCALE
O
x°
y°
62°
T
C
In the diagram, AC is a diameter of the circle centre O.
The tangents from T touch the circle at B and C.
t = 62° .
BCT
(a) Find x.
x = .................................................. [1]
(b) Find y.
y = ................................................ [2]
© UCLES 2020
4024/11/O/N/20
10
195 In this Venn diagram, shade the region M l , N , P .
!
M
N
P
[1]
20
(4x - 87) °
NOT TO
SCALE
2x°
(x + 60) °
y°
The diagram shows a cyclic quadrilateral.
Find the value of y.
y = ................................................ [4]
© UCLES 2020
0580/21/O/N/20
7
613
A
x°
B
NOT TO
SCALE
D
O
44°
C
A, B and C are points on a circle, centre O.
DA and DC are tangents.
Angle ADC = 44° .
Work out the value of x.
x = ................................................. [3]
© UCLES 2021
0580/22/M/J/21
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9
7
19 (a)
A
50°
T
P
B
NOT TO
SCALE
Q
P, Q and T are points on a circle.
ATB is a tangent to the circle at T and PT = PQ.
Find angle TPQ.
Angle TPQ = ..................................................... [2]
(b)
3x°
68°
2x°
NOT TO
SCALE
w°
The diagram shows a cyclic quadrilateral with an exterior angle of 68°.
Find the value of w and the value of x.
w = .....................................................
x = ..................................................... [3]
© UCLES 2021
0580/23/M/J/21
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Answer Key:
1. S18/qp11/q8 (E)
2. S18/qp12/q22 (E)
3. S20/qp21/q10 (0580) (E)
4. W20/qp11/q17 (E)
5. W20/qp21/q20 (0580) (E)
6. S21/qp22/q13 (0580) (E)
7. S21/qp23/q19 (0580) (E)
Difficulty level: Intermediate
19 1
C
B
y°
t°
z°
T
43°
A
x°
68°
O
D
In the diagram, A, B, C and D lie on the circle, centre O.
AD is a diameter.
The tangent to the circle at B meets the line DA produced at T.
t = 68° and CAO
t = 43°.
AOB
(a) Find x.
Answer x = ........................................ [1]
(b) Find y.
Answer y = ........................................ [1]
(c) Find z.
Answer z = ........................................ [1]
(d) Find t.
Answer t = ......................................... [1]
© UCLES 2015
4024/11/M/J/15
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17
2
24
B
C
y°
34°
t°
A
D
x°
O
z°
E
In the diagram, A, B, C, D and E lie on the circle, centre O.
BOE is a straight line.
ˆ = 34° .
DAB
(a) Find x.
Answer x = .................................... [1]
(b) Find y.
Answer y = .................................... [1]
(c) Find z.
Answer z = .................................... [1]
(d) Find t.
Answer t = .................................... [1]
© UCLES 2016
4024/12/O/N/16
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12
317
P
T
125º
O
X
S
Q
35º
R
In the diagram, P, Q, R, S and T lie on the circle.
QT is a diameter of the circle, centre O.
X is the point of intersection of PS and QT.
t = 125° and PStQ = 35°.
PXT
(a) Complete the following statement with a geometrical reason.
t = 35° because ..................................................................................................................... [1]
PTQ
t .
(b) Find PQT
t = .............................. [1]
Answer PQT
t .
(c) Find SPQ
t = ............................... [1]
Answer SPQ
t .
(d) Find SRQ
t = ............................... [1]
Answer SRQ
© UCLES 2017
4024/12/M/J/17
16
421
C
B
D
NOT TO
SCALE
y°
z°
x°
O
53°
t° E
A
In the diagram, the points A, B, C, D and E lie on the circle, centre O.
The points B, O and E lie on a straight line.
t = 53° .
AB is parallel to ED and DEO
(a) Find x.
x = ................................................... [1]
(b) Find y.
y = ................................................... [1]
(c) Find z.
z = ................................................... [1]
(d) Find t.
t = ................................................... [1]
© UCLES 2019
4024/11/O/N/19
15
520
A
B
NOT TO
SCALE
O
D
34° 69°
C
In the diagram, the points A, B, C, and D lie on the circle, centre O.
'
t = 69° and DCtA = 34° .
ACB
t .
(a) Find ABD
t = ................................................... [1]
ABD
t .
(b) Find AOD
t = ................................................... [1]
AOD
t .
(c) Find DAB
t = ................................................... [1]
DAB
© UCLES 2019
4024/12/O/N/19
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17
22
6
NOT TO
SCALE
D
E
54°
z° C
O y°
48°
x°
A
B
In the diagram, A, B, C, D and E lie on the circle, centre O.
t = 48° , DEB
t = 54° .
AOB
(a) Find x.
x = ................................................ [2]
(b) Find y.
y = ................................................ [1]
(c) Find z.
z = ................................................ [1]
© UCLES 2020
4024/12/O/N/20
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Answer Key:
1. S15/qp11/q19 (M)
2. W16/qp12/q24 (M)
3. S17/qp12/q17 (M)
4. W19/qp11/q21 (M)
5. W19/qp12/q20 (M)
6. W20/qp12/q22 (M)
Difficulty level: Hard
13
(a)
V
Q
P
U
R
38°
Circle I
Circle II
106°
S
T
The points P, T, U and V lie on Circle I, and the points P, Q, R, S and T lie on Circle II.
UPQ and VPR are straight lines.
VT̂ U = 38˚ and TŜR = 106˚.
Find
(i) VP̂ U,
[1]
(ii) QT̂ R,
[1]
(iii) TP̂ R,
[1]
(iv) UP̂ T.
[1]
(b)
A
y°
B
x°
O
C
A and B are points on the circle, centre O.
CA is a tangent at A and BOC is a straight line.
AĈ B = xº and BÂ O = yº.
Find an expression for y in terms of x.
© UCLES 2010
[3]
4024/21/O/N/10
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7
2
(b)
S
P
L
x°
Q
y°
R
M
TwocirclesintersectatLandM.
R and P are on the circumference of one circle. S and Q are on the circumference of the other
circle.
PLQ andRLSarestraightlines.
t =x°andMLQ
t = y°.
PLR
t =x°.
(i) Completetheproofthat SMQ
Statement
Reason
t =SLQ
t x°= PLR
.............................................................................................
t =SMQ
t =x°
SLQ
............................................................................................. [2]
t =y°.
(ii) Provethat PRM
Statement
Reason
[2]
(iii) Completethefollowingstatement,givingyourreasons.
ThetrianglesPRM andQSM are.............................................
Reasons......................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................
............................................................................................................................................... [3]
© UCLES 2015
4024/21/M/J/15
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19
3
(b)
B
A
60°
O
C
E
D
A, B, C, D and E are points on the circumference of a circle, centre O.
t = 60c.
AE = ED = DC = CB and AOB
t .
(i) Find ECD
Show all your working.
Answer
t = ........................... [2]
ECD
Answer
.................................. cm [2]
(ii) The radius of the circle is 12 cm.
Calculate the length of the minor arc AB.
Use r = 3.14 .
Question 25 is printed on the next page
© UCLES 2015
4024/11/O/N/15
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13
419
B
40°
A
P
R
Q
C
D
In the diagram, the two circles intersect at P and Q.
The line AB is a tangent to the circles at A and B.
AD and BC are diameters.
BD intersects the larger circle at R.
t = 40° .
DBC
t .
(a) Find CPR
t = ............................................. [1]
Answer CPR
t .
(b) Find CQR
t = ............................................. [1]
Answer CQR
t .
(c) Find ABD
t = ............................................. [1]
Answer ABD
t .
(d) Find ADB
t = ............................................. [1]
Answer ADB
© UCLES 2016
4024/11/M/J/16
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16
5
8
A
B
E
O
y°
D
F
X
G
C
The diagram shows two circles each with centre O.
A, B, C and D are points on the circumference of the large circle.
E, F and G are points on the circumference of the small circle.
CGD and CFB are tangents to the small circle.
Lines AEOC and FG intersect at 90° at X.
t = y° .
GOX
(a) Find each of these angles, as simply as possible, in terms of y.
Give reasons for your answers.
(i)
t
GEO
t = ............................ because ..............................................................................
Answer GEO
............................................................................................................................................... [2]
(ii)
t
GCX
t = ............................ because ..............................................................................
Answer GCX
............................................................................................................................................... [2]
(iii)
t
DAB
t = ............................ because ..............................................................................
Answer DAB
............................................................................................................................................... [2]
© UCLES 2017
4024/22/O/N/17
14
69
(a)
B
A
109°
C
NOT TO
SCALE
O
35°
28°
D
E
A, B, C, D and E lie on the circle, centre O.
Angle AEB = 35°, angle ODE = 28° and angle ACD = 109°.
(i)
Work out the following angles, giving reasons for your answers.
(a) Angle EBD = ............................. because ...........................................................................
..............................................................................................................................................
.............................................................................................................................................. [3]
(b) Angle EAD = ............................. because ...........................................................................
.............................................................................................................................................. [2]
(ii)
Work out angle BEO.
Angle BEO = ................................................ [3]
© UCLES 2018
0580/42/M/J/18
15
(b) In a regular polygon, the interior angle is 11 times the exterior angle.
(i)
Work out the number of sides of this polygon.
................................................. [3]
(ii)
Find the sum of the interior angles of this polygon.
................................................. [2]
© UCLES 2018
0580/42/M/J/18
14
7
18
C
B
x°
62°
127°
D
y°
A
z°
40°
E
In the diagram, A, B, C, D and E lie on the circle.
AB is parallel to ED.
t = 62°, CDE
t = 127° and BEC
t = 40°.
ABE
(a) Find x.
Answer x = .................................... [1]
(b) Find y.
Answer y = .................................... [1]
(c) Find z.
Answer z = .................................... [1]
© UCLES 2018
4024/12/O/N/18
9
8
13
D
C
NOT TO
SCALE
138°
O
12°
E
F
B
A
A, B, C, D and E are points on the circumference of a circle, centre O.
Tangent AF touches the circle at A.
O is the midpoint of AD.
t = 138° and BEO
t = 12° .
DOE
t .
(a) Find EAD
t = ................................................... [1]
EAD
t .
(b) Find EAF
t = ................................................... [1]
EAF
t .
(c) Find BEA
t = ................................................... [1]
BEA
t .
(d) Find BCD
t = ................................................... [2]
BCD
© UCLES 2019
4024/11/M/J/19
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7
9
20
D
C
NOT TO
SCALE
20°
O
131° B
T
A
A, B, C and D lie on the circle, centre O.
TA is a tangent to the circle at A.
Angle ABC = 131° and angle ADB = 20°.
Find
(a) angle ADC,
Angle ADC = ................................................. [1]
(b) angle AOC,
Angle AOC = ................................................. [1]
(c) angle BAT,
Angle BAT = ................................................. [1]
(d) angle OAB.
Angle OAB = ................................................. [1]
© UCLES 2020
0580/22/M/J/20
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Answer Key:
1. W10/qp21/q3(a) (H)
2. S15/qp21/q3(b) (H)
3. W15/qp11/q24(b)(i)(H)
4. S16/qp11/q19 (H)
5. W17/qp22/q8(a) (H)
6. S18/qp42/q9 (0580) (H)
7. W18/qp12/q18 (H)
8. S19/qp11/q13 (H)
9. S20/qp22/q20 (0580) (H)