Circle Theorem Circumscribed Circle

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 = 30.] [v = 60 ] . [w = 60 ], [x = 120,] [y = 40]

• [u = 60.] [v = 20 ] . [w = 50 ], [x = 100,] [y = 30]

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

Angle AOC = (3x+22)° and angle ABC = 5x°.

Find the value of x.

• 27

• 26

• 25

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

Angle ADC = 128°, angle ACD = 28° , angle BCO = 30° and angle AOC = 104°.Find angle BAO. .

• 22

• 23

• 40

A, B and C are points on the circle, centre O.

Find the obtuse angle AOC.

• 100

• 260

• 65

The points E, F and G lie on the circumference of a circle, centre O.

JGH is a tangent to the circle.

Find the value of x and the value of y.

• x=32 , y= 58

• x=29, y=58

• x=32 , y=32

The diagram shows the vertices of a triangle lying on the circumference of a circle with centre O.

Find the value of b.

Give a reason for your answer.

• b=90

  angle in a semicircle is 90°

• b=29

  angle in a semicircle is 90°

• b=30

  sum of interior angles of triangle is 180°

A, B and C are points on the circle, centre O.

AB and OC intersect at P.

Find the value of w.

• 55

• 21

• 73

A, B, C and D are points on the circumference on a circle.Find the size of angle DAB.

• 118

• 62

• 85

A, B and C are three points on the circumference of another circle.O is the centre of the circle.Angle AOB = 112°. Work out the size of angle ACB.

• 224

• 56

• 68

A, B, C and D are points on the circumference of a circle.Find x, the size of angle ACD. Give a reason for your answer.

• x= 50 same segment

• x=50 . angle in the semicircle is 90

• x=130, cyclic quadrilateral