1.
(a)
O is the centre of the circle.
A, B and C are points on the circumference.
Write down the value of angle x.
B
x
C
O
104°
A
Answer x = ........................................................degrees
(1)
(b)
P, Q and R are points on the circumference of the circle.
NPT is the tangent to the circle at P.
Q
Not drawn accurately
70°
z
N
R
52°
T
P
Calculate the value of z.
Give a reason for each step of your working.
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Answer .......................................................degrees
(3)
(Total 4 marks)
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2.
(a)
In the diagram, O is the centre of the circle.
Not drawn accurately
62°
a
O
Calculate the value of a.
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Answer ......................................................... degrees
(2)
(b)
In the diagram below, O is the centre of the circle and angle PSR = 100°.
Q
Not drawn accurately
R
O
b
P
100°
S
Calculate the value of b.
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Answer ......................................................... degrees
(2)
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(c)
CD is a tangent to the circle at C.
A
Not drawn accurately
c
B
50°
44°
D
C
Calculate the value of c.
Give reasons for your answer.
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Answer ......................................................... degrees
(3)
(Total 7 marks)
3.
(a)
O is the centre of the circle.
p
not drawn accurately
O
q
110°
(i)
Calculate the value of angle p.
Give a reason for your answer.
...........................................................................................................................
Answer p = ...........................................................................................degrees
Reason ...............................................................................................................
(2)
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(ii)
Calculate the value of angle q.
Give a reason for your answer.
Answer q = ...........................................................................................degrees
Reason ...............................................................................................................
(2)
(b)
UPT is a tangent to the circle.
QRS is a straight line.
S
R
not drawn accurately
Q
U
P
T
Prove that angle PRS = angle QPT.
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(3)
(Total 7 marks)
4.
(a)
Points P, Q, R and S lie on a circle.
PQ = QR
Angle PQR = 116°
Not drawn accurately
S
P
116º
R
Q
Explain why angle QSR = 32°.
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The Robert Smyth School
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(2)
(b)
The diagram shows a circle, centre O.
TA is a tangent to the circle at A.
Angle BAC = 58° and angle BAT = 74°.
B
Not drawn accurately
C
O
58º
74º
T
(i)
A
Calculate angle BOC.
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Answer Angle BOC = ............................... degrees
(1)
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(ii)
Calculate angle OCA.
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Answer Angle OCA = ............................... degrees
(3)
(Total 6 marks)
5.
ABCD is a cyclic quadrilateral.
PAQ is a tangent to the circle at A.
BC = CD
Angle QAB = 38° and angle BAD = 76°
Not drawn accurately
C
D
B
76°
P
38°
A
Q
Show that AD is parallel to BC.
Give reasons to justify any values you write down or calculate.
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The Robert Smyth School
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(Total 4 marks)
The Robert Smyth School
7
1.
(a)
52°
B1
(b)
52 at Q
M1
or angle NPQ = 70
may be credited from diagram
(angles in) alternate segment
B1
58
A1
58 as answer scores M1A1
[4]
2.
(a)
(b)
(c)
180 – 90 – 62 or 90 – 62
oe
M1
28
A1
Q = 80°
or reflex  POR = 200°
Note: 80° may be seen on diagram
M1
160
A1
 A = 44°
or third  at C = 86°
Allow 180 – 44 – 50
M1
M1
(z = ) 86
A1
‘Alternate segment’
oe
B1
[7]
3.
(a)
(i)
70
B1
(opposite angles of) cyclic quadrilateral
B1
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(ii)
140
B1
f.t. their (i)
angle at centre is twice angle at circumference
(b)
B1
 PRS = 180 –  PRQ
B1
 QPT = 180 –  QPU
B1
 PRQ =  QPU A1ternate segment
no mark unless alt segment stated
B1
[7]
4.
(a)
(b)
angle QPR = 32°
Base angle of isosceles triangle
B1
angle QSR = 32°
equal to angle QPR, angles in same segment oe
precise explanations for 2 marks
B1
(i)
116°
B1
(ii)
Line from O to A, creating 90°
Join OA
using alt seg thm
angle OAB = 16°
angle CA(X) = 48°
angle BCA = 74°
M1
M1
M1
angle OAC = 42°
angle OAC = 42°
angle BCO = 32°
M1
M1
M1
angle OCA = 42°
angle OCA = 42°
angle OCA = 42°
A1
A1
A1
[6]
5.
ABD = 66 (Alt segment)
or angles in triangle if ADB found first
B1
DCB = 104 (opposite in cyclic)
In all alternatives, for first 3 B marks do not award B1 the first
time no reason or wrong reason given, otherwise accept angles
identified in answer or on diagram.
NB Mark ‘positively’ ie, ignore wrong
values or
reasons unless totally contradictory.
B1
DBC = 38 (isosceles)
CBA = 104
B1
CBA + BAD = 180 (interior)
In all alternatives, reason must be given for final B1
Accept ‘allied’ or ‘angles between parallel lines’. Dependent on
correct angles.
B1
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Alt. 1 ADB = 38 (Alt segment)
B1
DCB = 104 (opposite in cyclic)
B1
CBD = 38 (isosceles)
B1
CBD = ADB (alternate)
B1
Use of ‘Z angles’ is not acceptable
Dependent on correct angles
Alt. 2 ADB = 38 (Alt segment)
DCB = 104 (opposite in cyclic)
B1
B1
BDC = 38 (isosceles)
ADC = 76
B1
BDC + BCD = 180 (interior)
Dependent on correct angles
B1
Alt. 3 ADB = 38 and ABD = 66 (Alt segment)
B1
DCB = 104 (opposite in cyclic)
B1
CBD = CBD = 38 (isosceles)
B1
DCB = CBA and CDA and BAD = (isosceles trapezium)
B1
[4]
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