In the diagram $PQ$ is parallel to $RS$.
$VW$ and $ZY$ are produced to meet at $X$.
$P\hat{V}Z={24}^o$, $R\hat{Y}Z={38}^o$ and $R\hat{W}V={71}^o$.

Find the value of angle $W\hat{X}Y$ and the value of reflex angle $V\hat{Z}Y$.

In the diagram, $A,B,C$ and $F$ lie on the circumference of the circle.
$AB$ is a diameter of the circle.
$AB$ and $DC$ produced meet at $E$.
$DB=BE$ and $A\hat{E}D={18}^o$.

(a) Find the value of angle $A\hat{B}C$.

(b) Show that $BD$ bisects angle $A\hat{B}C$.
Give reasons for each of your working.

In the triangle $ABC$, $AB$ is parallel to $DE$.
Angle $B\hat{A}D={110}^o$ and angle $A\hat{C}B={36}^o$.

(a) Find the size of the angle:
(i) $A\hat{B}C$,

(ii) $C\hat{D}E$,

(iii) $D\hat{E}B$.

(b) Explain why triangles $ABC$ and $DEC$ are similar.

The diagram shows a circle, centre O, a chord and tangent at the point T.
The chord has length 8 cm and is 3 cm from the centre O.
The tangent, PT, has length 12 cm.

Find the length OP.

The area of the sector is $12\pi$ cm².
Work out the length of the arc AB.
Give your answer in terms of $\pi$.

The diagram shows a sector of a circle with radius 5 cm and sector angle 36°.
Use $\pi =3.142$. Calculate the perimeter.

The diagram shows a circle centre O.
A, B, C, D, E are points on the circle and PQ is a tangent to the circle at A.
Angle ACD = 40° and angle PAE = 25°. Find the size of angle:
(a) AOD,

(b) ABE,

(c) AED.

A chord PQ is 3 cm away from the centre O, of a circle.
Another chord RS parallel to PQ is 4 cm away from the centre, O.
(a) Find the distance between the chords PQ and RS.

(b) State with a reason, which of the chords PQ or RS is longer.

The diagram shows points A, B, C, and D on the circumference of a circle centre O.
Angle BOD = 124°. Write the size of:
(a) Angle DCB,

(b) Angle DAB.

The points A, B, C and D are on the circumference of a circle.
Angle BAC = 27° and angle EBC = 70°.
Find:
(a) Angle BDC,

(b) Angle ACB,

(c) Angle ADC.

In the diagram A, B, C and D are points on the circumference of the circle.
AEC and DEB are straight lines.
AE = 4 cm, BC = 3 cm and AD = 9 cm. (a) Show that triangles AED and BEC are similar.

(b) Find the length of BE.

In the diagram, A, B, C and D are points on the circumference of the circle with centre O.
PCQ is a tangent to the circle at C and DQ is a tangent to the circle at D.
Angle ABC = 94° and angle BCP = 50°. (a) Find:
(i) Reflex angle AOC,

(ii) Angle BAO.

(b) If the area of triangle CDQ = 8 cm², find CD².

In the diagram, TA and TB are tangents to the circle at the points A and B.
TBS is a straight line.
The straight lines AB and QP intersect at Y.
Angle QAB = 27°, angle PQB = 40° and angle ATB = 56°. (a) State with reasons the values of the angles:
PAY,

(b) QBP,

(c) TAP,

(d) AYP.

In the diagram, three vertices L, M and N of a rhombus lie on the circumference of a circle centre O and radius 10 cm.
Angle MNO = 60°. (a) Show that angle LON = 120°.

(b) Calculate:
(i) The area of the rhombus,

(ii) The area of the shaded part.

(c) PQ is a tangent to the circle at Q.
PLN and NOQ are straight lines. (i) Find:
(a) Angle LPQ,

(b) Angle OLP,

(c) Angle LQP.

(ii) Show that $\Delta LPQ$, $\Delta NPQ$ and $\Delta LNQ$ are similar.

The diagram shows a right-angled triangle OPQ and a circle, centre O with radius r cm.
The circle cuts OP and OQ at A and B respectively.
AP = 7 cm, PQ = 5 cm, BQ = 6 cm and angle OQP = 90°. (a) Show that r = 6 cm.

(b) Find the value of $\sin POQ$.

(c) Calculate:
(i) The length of major arc AB,

(ii) The shaded area.

In the diagram AOC is the diameter of circle ABCD.
AT and DT are tangents to the circle.
BD = BA and angle DBA = 68°. Find the value of:
(a) w,

(b) x,

(c) y,

(d) z.

In the diagram, TB is a tangent to the circle at point B.
O is the centre of the circle.
The angle OAB = 50°, angle CAD = 49° and angle DCA = 25°. (a) Calculate the following angles and give a reason for each part.
(i) DBC

(ii) ADB

(iii) CBT

(b) Find:
(i) The angle CAO,

(ii) The angle OBC.

(a) In the diagram, JL and LN are chords of a circle equidistant from the centre.
M is the mid-point of JL. (i) Use accurate construction to find the centre of a circle and label it C.

(ii) State the relationship between JM and LN.
Give a reason.

(b) The diagram shows a circle with two equal chords VW and WX.
PR is a tangent to the circle at a point X.
Reflex angle VWX is 270°. Calculate the size of the marked angle WXR.

In the diagram, O is the centre of the circle and P, Q, R, S are points on the circumference.
RT is a tangent to the circle.
PS meets the tangent at T when produced.
Angle PQR = 103°, angle SOP = 120° and angle STR = 34°. (a) Find the value of:
(i) angle POR,

(ii) angle OST.

(b) Show that angle OPQ + angle ORQ = 103°.

(c) Explain why quadrilateral OSTR is not a trapezium.

In the diagram, the straight line ABC is parallel to GF and ED.
BF and CD are also parallel.
Angle ABF = 55°, angle DEF = 115° and angle BDC = 65°. (a) State, with reasons, the value of:
(i) Angle BCD,

(ii) Angle ABD,

(iii) Angle BFG.

(b) ED is produced to the point H such that DH = BC.
State with the reason the name of the quadrilateral BDHC.

The diagram shows the positions P, Q, R and S of four locations on an island.
P, S and R are on a straight line.
QS = 4 km, RS = 6 km, angle SPQ = 25°, angle PQS = 35° and Q is due East of P. Calculate:
(a) PQ,

(b) QR,

(c) The area of triangle QRS.