Circle Theorems (Cambridge O Level Maths)

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  and .

 
...................  ...................  ...................  ...................  ..................

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.

AT is a tangent to the circle at A.
 
Find the value of .
 

................................................

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 , , , and .

= ..................... , = ....................., = ..................... ,   

= ..................... , = .....................

Show that obtuse angle AOC = 104°.

Give a reason for each step of your working.

Find angle BAO.
 

Find angle ABD.

The diagram shows a circle with diameter .
is a tangent to the circle at .

Find the value of . 
 

= ..............................................

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

= .................................................     
= .................................................     
= .................................................     
= .................................................     

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

 Calculate angle .

Angle = ................................................ 

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 =  and angle ABC = .

Find the value of .

= ...................................................

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

Show that
You must give a reason for each stage of your working.

Dylan was asked to give some possible values for  and .

He said, " could be 200 and  could be 110, because 200 − 110 = 90"

Is Dylan correct?
You must give a reason for your answer.

 is a regular pentagon.
and  are points on a circle, centre .
and  are tangents to the circle.
is a straight line.

Prove that .