Circle Theorems (Cambridge (CIE) O Level Maths): Exam Questions

The diagram shows a cyclic quadrilateral $ABCD$.
$BD$and $AC$intersect at $X$.

Angle $BAD$= 74° and angle $BCA$= 34°.

Find

i) angle $BDA$

Angle $BDA$ = .............................................[1]

ii) angle $BCD$

Angle $BCD$ = ..............................................[1]

iii) angle $ABD$.

Angle $ABD$ = ..............................................[1]

In the diagram, triangle $ADX$is similar to triangle $BCX$.

$BC=4.5\mathrm{cm},AD=9\mathrm{cm}\mathrm{and}CX=3.3\mathrm{cm}.$

Work out $XD$.

$XD$ = ......................................... cm

In the diagram, K, L and M lie on the circle, centre O.
Angle KML = $2x°$and reflex angle KOL = $11x°$.

Find the value of $x$.   

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

$P$, $R$ and $Q$ are points on the circle.
$AB$ is a tangent to the circle at $Q$.
$QR$ bisects angle $PQB$ therefore
Angle $BQR=x°$  and $x<60$.

Use this information to show that triangle $PQR$ is an isosceles triangle.
Give a geometrical reason for each step of your work.

$A$, $B$, $C$ and $D$ lie on the circle.
$PCQ$ is a tangent to the circle at $C$.
Angle $ACQ$ = 64°.

Work out angle $ABC$, giving reasons for your answer.

Angle $ABC$ = .......................... because ..........................

$A,B,C$ and $D$ are points on the circumference of the circle.
The line $XY$ is a tangent to the circle at $A$.

Find the value of $x$, giving a reason for your answer.     

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

Find the value of $y$, giving a reason for your answer.   

$y$ = ............................. because .................................

In the diagram, A, B, C and D lie on the circumference of a circle, centre O.
Angle ACD = $x°$ and angle OAB = $2x°$.

Find an expression, in terms of $x$, in its simplest form for angle AOB .

Angle AOB = ................................................... 

Find an expression, in terms of $x$, in its simplest form for angle ACB.

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

Find an expression, in terms of $x$, in its simplest form for angle DAB.

Angle DAB = ...................................................

A, B and C are points on the circle, centre O. AB and OC intersect at P.  

Find the value of $w$.  

$w=$....................................................

$A,B,C$ and $D$ lie on the circle, centre $O$.
$TA$ is a tangent to the circle at $A$.
Angle $ABC=131°$ and angle $ADB=20°$.

Find angle $BAT$.  

Angle $BAT=$.................................................

Find angle $OAB$.  

Angle $OAB=$.................................................

$A,B,C$ and $D$ are points on the circumference of a circle.
$EF$ is a tangent to the circle at $A$.
Angle $BAE=43°$ and angle $ACD=59°$.
Chord $BC$ is parallel to chord $AD$.

Find angle $ABC$.
Give a geometrical property to explain each step of your working.

The diagram shows a cyclic quadrilateral.

Find the value of $y$.

$y=$ ................................................

$J$, $K$, $L$ and $M$ are points on the circumference of a circle with diameter $JL$. $JL$ and $KM$ intersect at $N$.
Angle $JNK=104°$ and angle $MLJ=22°$.

Work out the value of $d$.  

$d$ = ....................................................

$A,B,D$ 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.

$B,C,D$ 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.

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

$B,C,D$ 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.