Circle theorems I

A, B, C and D sit on the circumference of the circle, with $AC$ and $BD$ meeting at $X$.

Feb/March 2018

The points $E$, $F$, $G$ and $H$ all lie on the circle, with $EG = EH$. The lines $HF$ and $EG$ meet at $K$. $ET$ is tangent to the circle at $E$. Angle $FET$ is $47^{\circ}$ and angle $FEG$ is $25^{\circ}$.

Feb/March 2021

The diagram represents a circle with centre $O$ and diameter $AC$. The points $A, B, C, D$ and $E$ are all on the circle's circumference. This sketch is not drawn to scale. At $C$, the angle between $CB$ and $CO$ is $65^\circ$. At $D$, the angle between $DO$ and $DE$ is $46^\circ$. At $A$, the angle between $AO$ and $AB$ is labelled $x^\circ$. At $E$, the angle between $EO$ and $EA$ is labelled $y^\circ$.

Feb/March 2023

ABCD is a cyclic quadrilateral, ABX lies on a straight line and PQ is tangent to the circle at A. Angle CBX = 85^{\circ}, angle BAQ = 55^{\circ} and angle CAD = 42^{\circ}.

Feb/March 2023

A, B, C and D lie on a circle. ADX and BCX are straight lines. $\angle BAD = x^\circ$ and $\angle DCX = y^\circ$.

Feb/March 2024

A, B and C are points on a circle whose diameter is $AC$. The line segment $AC$ is extended to $D$, and $\angle BAC = 63^{\circ}$.

May/June 2015

The diagram depicts a circle with centre $O$ and diameter $AD$. Point $B$ lies on the circumference, and the straight line $CDE$ touches the circle at $D$. Also, $AD = 21$ cm and $CD = 16$ cm.

May/June 2015

In the diagram, $B, C, D$ and $E$ are on the circle with centre $O$. $AB$ and $AD$ touch the circle as tangents. Angle $BAD = 48\degree$. The diagram is NOT TO SCALE.

May/June 2015

A, B, P and Q are on a circle with centre $O$. The angle $APB$ is $56^{\circ}$. The diagram shows the circle with $A$, $B$, $P$ and $Q$ labelled. The angle at centre $O$ between $OA$ and $OB$ is marked $x^{\circ}$. The angle at $Q$ between $QA$ and $QB$ is marked $y^{\circ}$. Diagram not to scale.

May/June 2016

In the diagram, $A$, $B$ and $C$ are points on the circle’s circumference, with centre $O$. The diagram is shown NOT TO SCALE. The angle at $B$ between $OB$ and $AB$ is labelled $28^\circ$.

May/June 2016

The diagram depicts a cyclic quadrilateral $ABCE$. $AED$ and $BCD$ are straight lines. $AC=CD$, $\angle ABC=45^{\circ}$ and $\angle ACE=20^{\circ}$. The diagram is labelled NOT TO SCALE.

May/June 2016

The diagram depicts three straight lines intersecting at a single point. The angles labelled are $25^\circ$, $98^\circ$, $x^\circ$ and $y^\circ$.

May/June 2016

Points A, B, C, D and E lie on the circumference of a circle with centre $O$. $GAF$ is a tangent to the circle at $A$. $AB$ is parallel to $EC$ and $AB = AD$. The diagram is not drawn to scale.

May/June 2016

Points $A$, $B$ and $C$ lie on the circumference of a circle whose diameter is $AB$. A tangent has been drawn at $A$. The angle at $A$ between that tangent and line $AC$ is $42^\circ$. The diagram is NOT TO SCALE.

May/June 2017

Points $A$, $B$, $C$, $D$ and $E$ are on the circle. $AB$ is produced to $F$. Angle $AED = 140^\circ$ and angle $CBF = 95^\circ$. The figure labels angles $w^\circ$, $x^\circ$ and $y^\circ$. The diagram is not to scale.

May/June 2017

The diagram depicts a circle with centre $O$, and points $B$ and $D$ lie on the circumference. Line $AC$ is tangent to the circle at $B$. $OB$ is parallel to $DC$, and angle $OAB = 49^\circ$.

May/June 2017

A circle sketch, drawn not to scale, has points $A$, $B$, $C$, $D$ and $E$ placed on the circumference. The angles marked are $47\degree$ at $C$ (angle $DCE$) and $85\degree$ at $E$ (angle $CEA$). Inside the circle, the angles are labelled $w\degree$, $x\degree$ and $y\degree$.

May/June 2018

Points A, B and C lie on a circle with centre O. The angle at B between BA and BC is $130^{\circ}$. The diagram is labelled NOT TO SCALE.

May/June 2019

The diagram shows that $ABC$ is a straight line. $AD$ runs parallel to $BE$, $ ngle BAD = 34^\circ$ and $AB = BD$. The angles at the marked points are labelled $p^\circ$, $q^\circ$, $r^\circ$, $t^\circ$ and $s^\circ$. A second diagram presents a circle with centre $O$, points $B$ and $D$ on the circumference, line $AC$ touching the circle only at $B$, and $BD$ as a straight line passing through $O$.

May/June 2019

A, B and C are points on a circle with centre O. DA and DC are tangents, and angle ADC = $44^\circ$.

May/June 2021

P, Q and T lie on a circle. $ATB$ is a tangent to the circle at $T$, and $PT = PQ$.

May/June 2021

Points A, B and C are on a circle with centre O and AC as the diameter. The diagram is not drawn to scale. AB measures 6 cm and BC measures 10 cm.

May/June 2021

A, B, C and D lie on a circle. TU is a tangent to the circle at D, and DA is parallel to CB. The diagram marks angles at D of $38^\circ$ (between tangent TU and chord DC) and $60^\circ$ (inside the circle). Diagram NOT TO SCALE.

May/June 2022

ABC, DEF and GHK are triangles whose vertices all lie on the circumference of a circle. The angles shown are: at A, an interior angle of 6^{\circ}; at D, 8^{\circ}; at K, 85^{\circ}; at H, 85^{\circ}; at E, 94^{\circ}; and at F, 82^{\circ}. NOT TO SCALE.

May/June 2022

The diagrams are not drawn to scale.

May/June 2022

Work out the size of angle $PSQ$, and give a geometrical reason for every step in your working.

May/June 2022

The figure shows a circle with centre $O$ and diameter $AB$. Points $A$, $B$ and $C$ are on the circumference. The diagram is not drawn to scale.

May/June 2023

$A$, $B$, $C$ and $D$ lie on a circle. $AB$ is parallel to $DC$, and $\angle ACD = 32^{\circ}$. The chords $AC$ and $DB$ cross at $E$.

May/June 2023

The sketch shows a circle with centre O and points B, D and E on the circumference. AOEF lies on one straight line. The straight line AC is tangent to the circle at B. The angle marked at A is $28^\circ$.

May/June 2023

The diagram represents a cyclic quadrilateral.

May/June 2024

Points $P$, $Q$, $R$ and $T$ lie on a circle. $AB$ touches the circle at $T$. $ngle ATP=50^\circ$, $ngle PTR=48^\circ$ and $PQ=QR$. The diagram is not drawn to scale.

May/June 2024

Points $A$, $B$ and $C$ are on a circle with centre $O$. The diagram indicates an angle of $39^\circ$ at the circumference between the line $AB$ and the line $AO$. The diagram is not drawn to scale.

May/June 2025

The diagram depicts a circle with centre $O$. $P$ and $S$ lie on the circle. $POR$ is a straight line. $QRST$ is tangent to the circle at $S$. The marked angles are $25^{\circ}$ at $P$, $x^{\circ}$ at $S$, and $y^{\circ}$ at $R$ on the tangent.

May/June 2025

$P$, $Q$ and $R$ are points on a circle, and $QR$ is a diameter.

May/June 2025

Points A, B, C and D lie on a circle. EF touches the circle at A. AB is parallel to DC. The diagram shows angles 35^{\circ} and 60^{\circ} at A. Diagram not to scale.

May/June 2025

Points A, B and C lie on a circle with centre O and diameter AC. The diagram shows angle at A is 65^{\circ}, angle at B is x^{\circ}, and angle at C is y^{\circ}. It is labelled NOT TO SCALE.

May/June 2025

The diagram shows that $AP$ is a tangent to the circle at $P$. $O$ marks the centre of the circle, $\angle PAO = 37^{\circ}$ and $AP = 11\,\text{cm}$.

Oct/Nov 2015

Write down the mathematical name for the line $OD$.

Oct/Nov 2015

Points A, B, C, D and E lie on a circle with centre O. The angle $BAD$ is $37^{\circ}$.

Oct/Nov 2015

The diagrams are circles and are labelled NOT TO SCALE.

Oct/Nov 2015

In the diagram, $PT$ touches the circle at $P$ as a tangent. $PW$ is a diameter, and angle $TPQ = 42^{\circ}$.

Oct/Nov 2016

Points $A, B, C$ and $D$ are located on the circle with centre $O$.

Oct/Nov 2016

The figure contains four quadrilaterals, A, B, C and D. In quadrilateral A, the interior angles are 82^{\circ}, 70^{\circ}, 110^{\circ} and 98^{\circ}. In quadrilateral B, the interior angles are 113^{\circ}, 99^{\circ}, 67^{\circ} and 81^{\circ}. In quadrilateral C, the interior angles are 100^{\circ}, 57^{\circ}, 123^{\circ} and 80^{\circ}. In quadrilateral D, the interior angles are 96^{\circ}, 83^{\circ}, 97^{\circ} and 84^{\circ}. None of the diagrams is drawn to scale.

Oct/Nov 2016

A, B, C and D are located on the circle with centre O. BCE is a straight line. Angle AOC = 108^{\circ} and angle DCE = 60^{\circ}.

Oct/Nov 2017

The diagram depicts a circle with centre $O$. $A$, $B$ and $C$ lie on the circumference. $BC$ forms a diameter of the circle. $PQ$ is tangent to the circle at $C$, and $AOQ$ is a straight line. The angle shown at the centre is $48^\circ$.

Oct/Nov 2017

A circle with centre $O$ is shown. $AB$ and $DE$ are chords of the circle. $M$ is the midpoint of $AB$ and $N$ is the midpoint of $DE$. Also, $AB = DE = 9\text{ cm}$ and $OM = 5\text{ cm}$. The figure is labelled with the points $A, B, C, D, E, M, N$ and centre $O$, and it is marked NOT TO SCALE.

Oct/Nov 2018

Points A, B and C lie on the circumference of the circle, with centre O. The straight line DE is in contact with the circle at B. The diagram gives angle at A as 35^{\circ}.

Oct/Nov 2018

The diagram illustrates a circle with centre $O$. The straight line $ABC$ touches the circle at $B$ as a tangent. $OB = 8$ cm, $AB = 15$ cm and $BC = 22.4$ cm. $AO$ passes through the circle at $X$ and $OC$ passes through the circle at $Y$.

Oct/Nov 2018

In the figure, A, B, C and D are points on the circle with centre O. EA is a tangent to the circle at A, and angle EAB is $61^\circ$ while angle BAC is $55^\circ$.

Oct/Nov 2018

Points A, B and C lie on a circle with centre O. The diagram presents triangle ABC inside the circle, and the angle at B is shown with a right-angle marker. A straight line from A to C goes through O.

Oct/Nov 2019

On the circle centred at $O$, the points $A$, $B$, $C$ and $D$ are all located on the circumference. The given angles are $\angle ADC = 128^\circ$, $\angle ACD = 28^\circ$ and $\angle BCO = 30^\circ$. NOT TO SCALE.

Oct/Nov 2019

The figure represents a cyclic quadrilateral. The angles are marked as $(4x - 87)^\circ$, $(x + 60)^\circ$, $2x^\circ$ and $y^\circ$.

Oct/Nov 2020

Determine the length of this line in millimetres.

Oct/Nov 2020

Write down the mathematical term for line $PQ$.

Oct/Nov 2020

A, B, C and D lie on the circle with centre O. EF is tangent to the circle at D. Angle ADE = 42^{\circ} and angle COD = 162^{\circ}.

Oct/Nov 2020

Determine the value of $a$.

Oct/Nov 2021

Points A, B, C and D lie on a circle with centre O. The angle $COD$ is $124^{\circ}$, while $BCO$ is $35^{\circ}$. NOT TO SCALE.

Oct/Nov 2021

The figure depicts a quadrilateral whose interior angles are marked $101^\circ$, $95^\circ$, $85^\circ$ and $79^\circ$. It is labelled NOT TO SCALE.

Oct/Nov 2022

Find the value of $p$. Give a geometrical reason for your answer.

Oct/Nov 2022

Points $A$, $B$ and $C$ lie on the circle with centre $O$. The line $DE$ is tangent to the circle at $C$. $AC=10$ cm, $AB=9.5$ cm and $BC=7.7$ cm.

Oct/Nov 2022

A, B and C lie on a circle with centre O. NOT TO SCALE.

Oct/Nov 2023

Points A, B and C lie on the circumference of a circle, with centre O. The tangent DE meets the circle at C. Angle $BCE = 53^{\circ}$ and angle $ACO = 20^{\circ}$.

Oct/Nov 2023

Calculate the angle ACD.

Oct/Nov 2023

ABCDEF forms a regular hexagon, and $DF$, $DA$ and $DB$ are diagonals.

Oct/Nov 2023

Points A, B, C and D are on the circle. TAS is tangent to the circle at A. The diagram labels angles $37^\circ$, $41^\circ$, $x^\circ$ and $y^\circ$. The diagram is marked "NOT TO SCALE".

Oct/Nov 2024

State the mathematical name of this solid.

Oct/Nov 2024

B is a point on the circle with centre O. ABC is tangent to the circle at B, and angle OAB = 36^{\circ}.

Oct/Nov 2025

$A$, $B$, $C$, $D$ and $E$ are on the circle. $AC$ and $BD$ meet at $X$. Angle $ACD=55^{\circ}$ and angle $CXD=88^{\circ}$.

Oct/Nov 2025

Points A, B, C and D lie on the circumference of a circle with centre O. ED and EB are tangents to the circle, and AC is parallel to EB. Also, Angle AOD = 44^{\circ}.

Oct/Nov 2025

Calculate the measure of angle ATB.

Oct/Nov 2025

A, B and C are points on a circle with centre O. DE is a tangent to the circle at C. Angle ABC = $52^\circ$ and angle BCE = $65^\circ$.

Oct/Nov 2025