Circle theorems II

In the diagram, $EH$ runs parallel to $FG$. The lines $EG$ and $FH$ cross at $Z$. Angle $ZFG = 27^\circ$.

Feb/March 2017

A, B, C and D all sit on the circle. $PCQ$ is a tangent to the circle at C. Angle $ACQ$ is $64^\circ$.

Feb/March 2020

The points $A$, $B$, $C$, $D$ and $E$ all lie on one circle. $FG$ is tangent to the circle at $C$. $angle BAD = 110^\circ$, $angle ADB = 20^\circ$ and $angle BEC = 45^\circ$.

Feb/March 2025

$J, K, L$ and $M$ lie on the circumference of a circle whose diameter is $JL$. The lines $JL$ and $KM$ meet at $N$. Angle $JNK = 104^{\circ}$ and angle $MLJ = 22^{\circ}$. The diagram is not to scale.

May/June 2019

Diagram (a): On the upper horizontal line, the points are A and C; on the lower horizontal line, the points are F, B and D; E is positioned beneath B. The angle at A is $26^{\circ}$. Triangle ABC is isosceles. AC is parallel to FBD. CBE is a straight line. The angle at B on the right-hand side is labelled $x^{\circ}$. Not drawn to scale. Diagram (b): A circle has diameter $PQ$, with P at the top and Q at the bottom. The tangent $SPT$ touches the circle at P. The angles shown inside are $17^{\circ}$ and $58^{\circ}$. The angle at the interior point is labelled $y^{\circ}$. Not drawn to scale.

May/June 2019

Find $x$.

May/June 2023

Points A, B and C lie on a circle with centre O. DE is a tangent to the circle at A. Angle $ACO$ is $35^{\circ}$ and angle $BCO$ is $40^{\circ}$. The diagram is marked NOT TO SCALE.

May/June 2025

Here, $AP$ touches the circle at $P$, so $AP$ is a tangent. $O$ is the circle’s centre, $PAO = 37^\circ$ and $AP = 11$ cm. The diagram is not drawn to scale.

Oct/Nov 2015

In the circle shown, $A$, $B$, $C$ and $D$ all lie on the circumference, with centre $O$. Angle $ACD = x^{\circ}$ and angle $OAB = 2x^{\circ}$. The sketch is marked NOT TO SCALE.

Oct/Nov 2019

On the circle, $P$, $Q$ and $R$ lie on the circumference. $ST$ is tangent to the circle at $R$, and a second tangent from $S$ meets the circle at $V$. The diagram marks angles of $55^{\circ}$ at $P$, $65^{\circ}$ at $Q$, $60^{\circ}$ at $R$, together with $x^{\circ}$ between $QR$ and the tangent.

Oct/Nov 2021

This diagram depicts a circle with eight chords. The angles marked are $20^{\circ}$ and $72^{\circ}$ at the upper-right point on the circumference, $56^{\circ}$ at an internal intersection, $u^{\circ}$ at the lower-right point, $w^{\circ}$ at the lowest point, and $x^{\circ}$ and $v^{\circ}$ at the left point.

Oct/Nov 2022

For part (a), a circle with centre $O$ is drawn, and $A$, $B$ and $C$ are points on it. The angle $OBA$ is labelled $28^{\circ}$. For part (b), $P$, $Q$ and $R$ are points on a circle. $TU$ is a tangent to the circle at $P$. The angle $TPR$ is $47^{\circ}$ and the angle $PRQ$ is $52^{\circ}$. Both diagrams are marked NOT TO SCALE.

Oct/Nov 2023

Points $A$, $B$ and $C$ lie on the circumference of a circle with centre $O$. $DA$ and $DC$ are tangents to the circle. Angle $ABC = 64^\circ$. NOT TO SCALE.

Oct/Nov 2024

The diagram depicts a cyclic quadrilateral, and the ratio is $p:m=2:3$.

Oct/Nov 2025