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# Form 3 Mathematics Angle Properties of a Circle Questions and Answers

In these online lessons, we are going to solve several topical questions on angle properties of a circle, which is a form 3 mathematics topic.

Lessons (**29**) * SHARE*

- 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.

1m 27s - 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

4m 3s - 3.
In the figure below (not drawn accurately) PAQ is a tangent to the circle at A. Find the #angle#DAB and #angle#BAQ.

3m 53s - 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.

2m 21s - 5.
In the figure below, ABC is the tangent to the circle at B. #angle#ABG=#42^0# and #angle#FEG=#20^0#. (a) Find
(i) #angle#BGF
(ii) #angle#BDF
(b) If BE bisects #angle#GED, find #angle#DBC

6m 50s - 6.
In the figure below, O is the centre of the circle and #angle# OAC=#38^0#. Find #angle# ABC.

2m 5s - 7.
The figure below (not drawn to scale) shows a circle PQRS centre O with SR produced to T. PQ//SR and #angle# QSR=#35^0#. Calculate the size of #angle# QRT

2m 27s - 8.
In the figure below, #angle# CAD=#20^0#, #angle# AFE=#120^0# and BCDF is a cyclic quadrilateral. Find #angle# FED.

3m 32s - 9.
In the figure besides, CP=CQ and #angle# CQR=#160^0#. If ABCD is a cyclic quadrilateral, find #angle# BAD.

1m 44s - 10.
In the figure below, ABCD is a cyclic quadrilateral and BD is a diagonal. EADF is a straight line, #angle# CDF=#68^0#,#angle# BDC=#45^0# and #angle# BAE=#98^0#.
Calculate the size of
(a)# angle#ABD
(b)#angle# CBD

3m 35s - 11.
In the figure below AOC is a diameter of the circle centre O, AB=BC and #angle# 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) #angle# BAD
(ii) The obtuse #angle# BOD
(iii) #angle# BGD
(b) Show that #angle#ABE=#angle# CBF

8m 58s - 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

6m 24s - 13.
In the figure below, QOT is a diameter, #angle# QTP=#48^0#, #angle# TQR=#76^0# and #angle# SRT=#37^0#.
Calculate
(a) #angle# RST
(b) #angle# SUT
(c) Obtuse #angle# ROT
(d) #angle# PST

10m 22s - 14.
ABCD is a cyclic quadrilateral and AB is a diameter. Angle ADC=#117^0#. Giving reasons for each step, calculate #angle# BAC

4m 57s - 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. #angle# BPQ=#80^0#, #angle# PBU=#115^0# and #angle# BUS=#70^0#. Find the values of the following angles, stating your reasons in each case.
(a) #angle# BAR
(b) #angle# STR
(c) #angle# BSU
(d) #angle# BRS

6m 10s - 16.
On the figure below lines ABC and DC are tangents to the circle at B and D respectively. #angle# ACD=#40^0# and #angle# ABE=#60^0#.
Giving reasons find the size of
(a) CBD
(b) CDE

3m 51s - 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, #angle# DEC=#28^0# and #angle#AGE=#32^0#.
Calculate:
(a) #angle# AEG
(b) #angle# ABC

2m 42s - 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.

2m 55s - 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) #angle# MLN
(b) #angle# OLN
(c) #angle# LNP
(d) #angle# MPN

10m 14s - 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 #angle# QPR=#20^0#.
Find the size of #angle# RST

3m 16s - 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

2m 44s - 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, #angle #RST=#50^0# and #angle# RTV=#150^0#.
(a) Calculate the size of:
(i) #angle# ORS
(ii) #angle# USP
(iii) #angle# 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

8m 50s - 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

1m 26s - 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) #angle#QPR
(b) #angle# PQS
(c) #angle# OQS
(d) #angle# RTS
(e) #angle# RSV

7m 55s - 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

2m 39s - 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.

2m 9s - 27.
In the figure below, PQRS is a cyclic quadrilateral. PQ=QR, #angle# PQR=#105^0# and PS is parallel to QR.
Determine the size of:
(a) #angle# PSR
(b) #angle# PQS

2m 56s - 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.

2m 5s - 29.
In the figure below, O is the centre of the circle, CO is the parallel to BA and #angle# AOB =#96^0#. Calculate #angle# CAO

2m 58s