Circle Theorems

and  are points on the circumference of a circle, centre .
 is a tangent to the circle.
 is a straight line.

Angle 

Work out the size of the angle marked .
Give reasons for your answer.

and  are points on the circumference of a circle, centre .
 is a diameter of the circle.
and  are tangents to the circle.
Angle .

Work out the size of angle .

and  are points on the circumference of a circle, centre .

Angle .

Find the size of angle  in terms of .
Give a reason for each stage of your working.

and  are points on the circumference of a circle, centre .
 is a straight line.
Angle 

Find the size of angle .
Give a reason for each stage in your working.

The points  and  lie on a circle.
 is a straight line.

Angle 

Work out the size of angle .
You must give a reason for each stage of your working.

APB and CPD are chords of a circle.

AP = 14 cm            PB = 6cm             CP = 10 cm

Calculate the length of PD.

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

ED = 24 cm                       DC = 15 cm                       BC = 18 cm

Work out the length of AB.

and are points on a circle, centre

is a diameter of the circle.

Angle

Calculate the size of angle
Give a reason for each stage of your working.

Angle DFG = ....................................................... 

is a diameter of a circle.
is a chord of the circle.

Calculate the radius of the circle.

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

and are chords of a circle.

= 9 cm  = 6 cm = 8 cm

Calculate the length of .

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

is a quadrilateral.

Prove that   is not a cyclic quadrilateral.

and are points on a circle.

and are tangents.

Work out the size of angle .

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

and are points on a circle.

.

Is the centre of the circle?
Tick a box.

Yes             No

Show working to support your answer.

,  and  are points on a circle.

 is a straight line.

 is a tangent to the circle.

Sam is trying to work out the size of angle .
Here is his working.

angles in the same segment are equal angles at a point on a straight line add up to 180°

Make a criticism of his working.

and  are points on a circle.

 is a tangent to the circle.

Write down the size of angle .
Give a reason for your answer.

............................degrees 

and are points on a circle.

Circle the line that is a diameter of the circle.

In the diagram,

• E, F and D are points on the circumference of the circle
• G, D and H lie on a tangent to the circle
• angle EFD = 55°
• angle FDE = 43°.

Explain why angle HDF is 82°.

The diagram shows a circle, centre O.

Points A, B, C and D lie on the circumference of the circle.
EDF is a tangent to the circle.
Angle ABC = 82° and angle ODC = 57°.

Work out the value of x.

x = ......................

Work out the value of y.

y = ..................

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

PQ is a tangent to the circle at D.
Angle BDQ = 72° and angle ABD = 63°.

Work out angle x.
Give a reason for your answer.

^{Angle x = ...................° because ......................}

Work out angle y.
Give a reason for your answer.

Angle y = ...................° because .....................

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

Angle BAD = 52°.

Work out angle x.
Give a reason for your answer.

x = ................................... ° reason ..................................................................................

Work out angle y.
Give a reason for your answer.

y = ................................... ° reason ..................................................................................