Circle Theorems (Cambridge International IGCSE Mathematics): Exam Questions

In the diagram, A, B, C and D lie on the circle, centre O. EA is a tangent to the circle at A. Angle EAB = 61° and angle* BAC* = 55°.

Find angle BAO.

Angle BAO = ................................................

Extract:

In the diagram, A, B, C and D lie on the circle, centre O. EA is a tangent to the circle at A. Angle EAB = 61° and angle BAC = 55°.

Find angle AOC.

Angle AOC = ................................................

Extract:

In the diagram, A, B, C and D lie on the circle, centre O. EA is a tangent to the circle at A. Angle EAB = 61° and angle BAC = 55°.

Find angle ABC.

Angle ABC = ................................................

Extract:

In the diagram, A, B, C and D lie on the circle, centre O. EA is a tangent to the circle at A. Angle EAB = 61° and angle BAC = 55°.

Find angle CDA.

Angle CDA = ................................................

A, B, C and D are points on the circumference of the circle, centre O. DOB is a straight line and angle DAC = 58°.

Find angle CDB.

Angle CDB = ...............................................

P, Q and R are points on the circumference of the circle, centre O. PO is parallel to QR and angle *POQ *= 48°.

Find angle OPR.

Angle OPR = ..............................................

A, B, C, D and E lie on the circle, centre O. Angle AEB = 35°, angle ODE = 28° and angle ACD = 109°.

Work out the following angles, giving reasons for your answers.

i) Angle EBD = ............................. because .............................

ii) Angle EAD = ............................. because .............................

Extract:

A, B, C, D and E lie on the circle, centre O. Angle AEB = 35°, angle ODE = 28° and angle ACD = 109°.

Work out angle BEO.

Angle BEO = ................................................

K, L and M are points on the circle. KS is a tangent to the circle at K. KM is a diameter and triangle KLM is isosceles.

Find the value of $z$.

$z=$...............................................

F, G, H and J are points on the circle. EFG is a straight line parallel to* JH*.

Find the value of $y$.

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

The points A, B, C, D and E lie on the circle. PAQ is a tangent to the circle at *A *and EC = EB. Angle ECB = 80° and angle ABE = 40°.

Find the values of $v,w,x,y$ and $z$.

$v=$ ................... $w=$................... $x=$................... $y=$................... $z=$ ..................

A, B, *C *and D are points on the circle, centre O. *EF *is a tangent to the circle at D. Angle ADE = 42° and angle COD = 162°.

Find the following angles, giving reasons for each of your answers.

i) Angle $x$

$x=$ ............................ because ............................ [2]

ii) Angle $y$

$y=$ ............................ because ........................... [2]

iii) Angle $z$

$z=$ ............................ because ........................... [3]

AT is a tangent to the circle at A.

Find the value of $x$.

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

A, B, C and D lie on the circle, centre O, with diameter AC. PQ is a tangent to the circle at A. Angle PAD = 60° and angle BAC = 20°.

Find the values of $u$, $v$, $w$, $x$ and $y$.

$u$ = ..................... , $v$ = ....................., $w$ = ..................... , $x$ = ..................... , $y$ = .....................

In the diagram, A, B, C and D lie on the circle, centre O. Angle ADC = 128°, angle ACD = 28° and angle BCO = 30°.

Show that obtuse angle AOC = 104°.

Give a reason for each step of your working.

Extract:

In the diagram, A, B, C and D lie on the circle, centre O.

Angle ADC = 128°, angle ACD = 28° and angle BCO = 30°.

i) Find angle BAO.

Angle BAO = ................................................... [2]

ii) Find angle ABD.

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

The diagram shows a circle with diameter $PQ$. $SPT$ is a tangent to the circle at $P$.

Find the value of $y$.

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

P, Q, R, S and T lie on the circle, centre O. Angle PST = 75° and angle QTS = 85°.

Find the values of $v$, $w$, $x$ and $y$.

$v$ = .................................................
$w$ = .................................................
$x$ = .................................................
$y$ = .................................................

A, B, C and D are points on the circle, centre O. Angle ABD = 21° and CD = 12 cm.

Calculate the area of the circle.

.......................................... cm²

The diagonals of the cyclic quadrilateral ABCD intersect at X.

Explain why triangle ADX is similar to triangle BCX. Give a reason for each statement you make.

A, B and C lie on the circle, centre O. Angle AOC = $(3x+22)^{\circ}$ and angle ABC = $5x^{\circ}$.

Find the value of $x$.

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

$A,B,C$ 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.

Extract:

Show that y - x = 90

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.

$PQRST$ 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 $ST=UT$.