Circle Theorems (Cambridge (CIE) IGCSE Maths: Extended)

The diagram shows a cyclic quadrilateral .
and intersect at .

Angle = 74° and angle = 34°.

Find

i) angle 

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

ii)angle 

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

iii)angle .

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

In the diagram, triangle is similar to triangle .

Work out .

 = ......................................... cm

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

Find the value of .   

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

,  and  are points on the circle.
is a tangent to the circle at .
bisects angle therefore
Angle   and .

Use this information to show that triangle is an isosceles triangle.

Give a geometrical reason for each step of your work.

, ,  and  lie on the circle.
 is a tangent to the circle at .
Angle  = 64°.

Work out angle , giving reasons for your answer.

Angle = .......................... because ..........................

 and are points on the circumference of the circle.
The line is a tangent to the circle at .

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

= ............................. because .................................

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

= ............................. because .................................

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

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

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

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

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

Find an expression, in terms of , 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 .  

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

and  lie on the circle, centre . is a tangent to the circle at .
Angle  and angle .

Find angle .  

Angle .................................................

Find angle .  

Angle .................................................

The diagram shows a cyclic quadrilateral.

Find the value of .

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

, , and  are points on the circumference of a circle with diameter .
and  intersect at .
Angle  and angle .

Work out the value of .  

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

and  are points on a circle.
and  are straight lines.

Prove that triangle  is similar to triangle .
You must give reasons for your working.

and  are points on a circle.
, , and are straight lines.

Work out the size of angle .
Show your working clearly.

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

and  are points on a circle.

is the tangent at  to the circle.
is parallel to .

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