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Form 3 Mathematics Angle Properties of a Circle Questions and Answers
In the figure besides, CP=CQ and
∠
CQR=
160
0
. If ABCD is a cyclic quadrilateral, find
∠
BAD.
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1.
In the figure below, O is the centre of the circle. Angle OAB =
30
0
and angle BAC =
23
0
. Find the angle ACB.
2.
In the figure below, O is the centre of the circle. PQ and PR are tangents. Angle PQS=
40
0
and angle PRS=
30
0
Find angle (i) RTQ (ii) ORQ (iii) RPQ
3.
In the figure below (not drawn accurately) PAQ is a tangent to the circle at A. Find the
∠
DAB and
∠
BAQ.
4.
In the figure alongside O is the centre of the circle. Angle BAC =
50
0
and angle ABO=
20
0
. Determine the size of angle ACB.
5.
In the figure below, ABC is the tangent to the circle at B.
∠
ABG=
42
0
and
∠
FEG=
20
0
. (a) Find (i)
∠
BGF (ii)
∠
BDF (b) If BE bisects
∠
GED, find
∠
DBC
6.
In the figure below, O is the centre of the circle and
∠
OAC=
38
0
. Find
∠
ABC.
7.
The figure below (not drawn to scale) shows a circle PQRS centre O with SR produced to T. PQ//SR and
∠
QSR=
35
0
. Calculate the size of
∠
QRT
8.
In the figure below,
∠
CAD=
20
0
,
∠
AFE=
120
0
and BCDF is a cyclic quadrilateral. Find
∠
FED.
9.
In the figure besides, CP=CQ and
∠
CQR=
160
0
. If ABCD is a cyclic quadrilateral, find
∠
BAD.
10.
In the figure below, ABCD is a cyclic quadrilateral and BD is a diagonal. EADF is a straight line,
∠
CDF=
68
0
,
∠
BDC=
45
0
and
∠
BAE=
98
0
. Calculate the size of (a)
∠
ABD (b)
∠
CBD
11.
In the figure below AOC is a diameter of the circle centre O, AB=BC and
∠
ACD=
25
0
, EBF is a tangent to the circle at B. G is the point on the minor arc CD. (a) Calculate the size (i)
∠
BAD (ii) The obtuse
∠
BOD (iii)
∠
BGD (b) Show that
∠
ABE=
∠
CBF
12.
In the figure below, PQR is the tangent to the circle at Q. TS is a diameter and TSR and QUV are straight lines. QS is parallel to TV. Angle SQR=
40
0
and angle TQV=
55
0
. Find the following angles, giving reasons for each answer: (a) QTS (b) QRS (c) QVT (d) UTV
13.
In the figure below, QOT is a diameter,
∠
QTP=
48
0
,
∠
TQR=
76
0
and
∠
SRT=
37
0
. Calculate (a)
∠
RST (b)
∠
SUT (c) Obtuse
∠
ROT (d)
∠
PST
14.
ABCD is a cyclic quadrilateral and AB is a diameter. Angle ADC=
117
0
. Giving reasons for each step, calculate
∠
BAC
15.
The figure below shows two circles ABPQ and ABSR intersecting at A and B. PBS, QART and ABU are straight lines. The line UST is a tangent to the circle ABSR at S.
∠
BPQ=
80
0
,
∠
PBU=
115
0
and
∠
BUS=
70
0
. Find the values of the following angles, stating your reasons in each case. (a)
∠
BAR (b)
∠
STR (c)
∠
BSU (d)
∠
BRS
16.
On the figure below lines ABC and DC are tangents to the circle at B and D respectively.
∠
ACD=
40
0
and
∠
ABE=
60
0
. Giving reasons find the size of (a) CBD (b) CDE
17.
The diagram below shows a circle ABCDE. The line FEG is a tangent to the circle at point E. Line DE is parallel to CG,
∠
DEC=
28
0
and
∠
AGE=
32
0
. Calculate: (a)
∠
AEG (b)
∠
ABC
18.
In the figure below, O is the centre of the circle ABCD and AOD is a straight line. If AB=BC and angle DAC=
40
0
, calculate angle BAC.
19.
In the figure below, K,L,M and M are points on the circumference of a circle centre O. The points K,O,M and P are on a straight line. PN is a tangent to the circle at N. Angle KOL=
130
0
and angle MKN=
40
0
. Find the values of the following angles, stating the reasons in each case: (a)
∠
MLN (b)
∠
OLN (c)
∠
LNP (d)
∠
MPN
20.
In the figure below R,T and S are points on a circle centre O. PQ is tangent to the circle at T, POR is a straight line and
∠
QPR=
20
0
. Find the size of
∠
RST
21.
In the figure below, O is the centre of the circle passes through the points T, C and D. Line TC is parallel to OD and line ATB is a tangent to the circle at T. Angle DOC=
36
0
. Calculate the size of angle CTB
22.
In the figure below P, Q, R and S are points on the circle centre O. PRT and USTV are straight lines. Line UV is a tangent to the circle at S,
∠
RST=
50
0
and
∠
RTV=
150
0
. (a) Calculate the size of: (i)
∠
ORS (ii)
∠
USP (iii)
∠
PQR (b) Given that RT=7 cm and ST=9 cm, calculate to 3 significant figures: (i)the length of line PR (ii)the radius of the circle
23.
In the figure below, ABCD is a cyclic quadrilateral. Point O is the centre of the circle. Angle ABO =
30
0
and angle ADO =
40
0
. Calculate the size of angle BCD
24.
In the figure below, PR is a diameter of the circle centre O. Points P,Q, R and S on the circumference of the circle. Angle PRQ=
72
0
, QS=QP and line USV is a tangent to the circle at S Giving reasons, calculate the size of: (a)
∠
QPR (b)
∠
PQS (c)
∠
OQS (d)
∠
RTS (e)
∠
RSV
25.
In the figure below, BOD is the diameter of the circle centre O. Angle ABD =
30
0
and angle AXD=
70
0
. Determine the size of: (a)reflex angle BOC (b)angle ACO
26.
In the figure below, O is the centre of the circle. A, B, C and D are points on the circumference of the circle. Line AB is parallel to line DC and angle ADC=
55
0
. Determine the size of angle ACB.
27.
In the figure below, PQRS is a cyclic quadrilateral. PQ=QR,
∠
PQR=
105
0
and PS is parallel to QR. Determine the size of: (a)
∠
PSR (b)
∠
PQS
28.
An arc of a circle subtends an angle of
150
0
at the circumference of the circle. Calculate the angle subtended by the same arc at the centre of the circle.
29.
In the figure below, O is the centre of the circle, CO is the parallel to BA and
∠
AOB =
96
0
. Calculate
∠
CAO
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