Circle Theorems (AQA GCSE Maths: Higher): Exam Questions

$B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$ABE$ and $ADF$ are tangents to the circle.

Angle $DAB=40°$
Angle $CBE=75°$

Work out the size of angle $ODC$.

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$AC$ is a diameter of the circle.
$AC$ and $BD$ intersect at E.

Angle $CAB=25°$
Angle $DEC=100°$

Work out the size of angle $DAC$.
You must show all your working.

$P,M$ and $S$ are points on a circle, centre $O$.
$RST$ is a tangent to the circle.

Angle $PSO=48°$
$MP=MS$

Work out the size of angle $MST$.
Give reasons for each stage of your working.

$B,C$ and $D$ are points on the circumference of a circle, centre $O$. $ABO$ is a straight line.
$AD$ is the tangent at $D$ to the circle.
Angle $DAO=40°$

Work out the size of angle $BCD$.
Give a reason for each stage of your working.

$A$ and $B$ are points on a circle, centre $O$.

$BC$ is a tangent to the circle.
$AOC$ is a straight line.
Angle $ABO=x°$.

Find the size of angle $ACB$, in terms of $x$.
Give your answer in its simplest form.
Give reasons for each stage of your working.

$AOC$ and $BOD$ are diameters of a circle, centre $O$.

Prove that triangle $ABD$ and triangle $DCA$ are congruent.

$A,B,C$ and $D$ are four points on the circumference of a circle.

$AEC$ and $BED$ are straight lines.

Prove that triangle $ABE$ and triangle $DCE$ are similar.
You must give reasons for each stage of your working.

$P,Q,R$ and $S$ are points on a circle.

$PXR$and $QXS$are straight lines.
$PX=SX$

Prove that $QS$ is not a diameter of the circle.

$A,B$and $C$ are points on a circle.

• $BC$ bisects angle $ABQ$.

• $PBQ$ is a tangent to the circle.

Angle $CBQ=x$
Prove that $AC=BC$

$B,D,E$ and $F$ are points on a circle.
$ABC$ is the tangent at $B$ to the circle.
Angle $ABD=39°$
Angle $EFD=18°$

Work out the size of angle $BDE$.
Give reasons for your working.

In the diagram,

• A, B and C are points on the circumference of a circle

• O is the centre of the circle

• angle OAB = 15°

• angle BCO = 10°.

Calculate the acute angle AOC.

A and B are points on the circumference of a circle, centre O.
CA and CB are tangents to the circle.

Prove that triangle OAC is congruent to triangle OBC.

A, B, C and D are points on the circumference of a circle, centre O.
AC is a diameter of the circle.
Angle ABD = 58°.
Angle CDB = 22°.

Work out the sizes of angle ACD and ACB, giving reasons for your answers.

Angle ACD = ...........................°

Angle ACB = ...........................°

The diagram below shows a circle with centre $O$ and chords $AB$, $AC$, $BD$ and $CD$.

Chords $BD$ and $AC$ intersect at the point $E$.

$\angle CED$ is $101°$ and reflex $\angle AOD$ is $279°$.

Work out the size of $\angle BDC$. Give reasons for your working.

$A,B,D$ and $E$ are points on a circle.
$ABC$ and $EDC$ are straight lines.

Prove that triangle $BCD$ is similar to triangle $ECA$.
You must give reasons for your working

$A,B,C$ and $D$ are points on the circumference of a circle, centre $O$.
$FDE$ is a tangent to the circle.

Show that $y-x=90$
You must give a reason for each stage of your working.

Dylan was asked to give some possible values for $x$ and $y$.

He said, "$y$ could be 200 and $x$ could be 110, because 200 − 110 = 90"

Is Dylan correct?
You must give a reason for your answer.

$A,B$ and $C$ are points on the circumference of a circle centre $O$.

Prove that angle $BOC$ is twice the size of angle $BAC$.

$A,B$ and $C$ are points on the circumference of a circle, centre $O$. $AOB$ is a diameter of the circle.

Prove that angle $ACB$ is $90°$
You must not use any circle theorems in your proof.

$PQRST$ is a regular pentagon.
$R,U$ and $T$ are points on a circle, centre $O$.
$QR$ and $PT$ are tangents to the circle.
$RSU$ is a straight line.

Prove that $ST=UT$.

$C,D$ and $E$ are points on a circle.

$CE=DE$
The tangent at $D$ is shown.
$ACD$ and $BCE$ are straight lines.

Prove that   $y=3x$

$A,B$ and $C$ are points on a circle.

$CD$ is a tangent.

Assume that triangle $ABC$ is isosceles with $AC=BC$

Prove that $AB$ is parallel to $DC$.

In fact, triangle $ABC$ is equilateral.

Tick the two boxes for the statements that must be correct. 

Here is a cyclic quadrilateral.

$x:y=5:7$

Work out the size of angle $w$.

.....................degrees

$A,B,C,D$ and $E$ are points on a circle.

$BFD$ and $AFC$ are straight lines.

$DC=DF$

Work out the size of angle$x$.

You must show your working which may be on the diagram.

..........................degrees

$A,B$ and $C$ are three points on the circumference of a circle, centre $O$.

$BD$ and $CD$ are tangents to the circle.
$ABDC$ is a kite.
Angle $BDC$ is $x$

Prove that angle $ABO$is  $45°–\frac{x}{4}$

$A,B,C$ and $D$ are points on a circle.
$PCQ$ is a tangent to the circle.
$AB=CB.$

Angle $BCQ=x°$

Prove that angle $CDA=2x°$
Give reasons for each stage in your working.

$B,C,D$ and $E$ are points on a circle.

$AB$ is the tangent at $B$ to the circle.
$AB$ is parallel to $ED$.
$\mathrm{Angle}ABE=73°$

Work out the size of angle $DCE$.
Give a reason for each stage of your working.

$B,C,D$ and $F$ are points on a circle.
$ABC$, $AFD$, $BFE$and $CDE$are straight lines.
Work out the size of angle $x$.
Show your working clearly.

$x$ = ......................

$A,B$ and $C$ are points on a circle with centre $O$.

Angle $ABC=75°$

The area of the shaded segment is 200cm²

Calculate the radius of the circle.
Give your answer correct to 3 significant figures.

...................................................... cm

A, B, C and D are points on the circumference of a circle, centre O.

Angle BAD = 112° and angle DCO = 33°.

Show that angle y = 35°.
Give reasons for each stage of your working.

Work out angle z.

Give reasons for your answer.

Angle z = ......................... ° because ........................................................................................

A, B, C and D are points on the circumference of a circle, centre O.

Angle CAD = 28° and CD = 6.4cm.
BD is a diameter of the circle.

Calculate the area of the circle.

...................... cm²