The circle theorems

Exercise 1

Question 1

Diagram NOT accurately drawn

In the diagram, A, B, C and D are points on the circumference of a circle, centre O. Angle BAD = 70°. Angle BOD = x°. Angle BCD = y°.

(a) Work out the value of x. Give a reason for your answer.

(b) Work out the value of y. Give a reason for your answer.

Question 2

P, Q, R and S are points on the circumference of a circle, centre O. PR is a diameter of the circle. Angle PSQ = 56°.

(a) Find the size of angle PQR. Give a reason for your answer.

(b) Find the size of angle PRQ. Give a reason for your answer.

(c) Find the size of angle POQ. Give a reason for your answer.

Question 3

The diagram shows a circle, centre O. AC is a diameter. Angle BAC = 35°.
D is the point on AC such that angle BDA is a right angle.

(a) Work out the size of angle BCA. Give reasons for your answer.

(b) Calculate the size of angle DBC.

(c) Calculate the size of angle BOA.

Question 4

A, B, C and D are points on the circumference of a circle, centre O. BOD is a straight line. Angle COD = 70°

(b) Find the size of angle CBD. Give a reason for your answer.

Question 5

A, B, C and D are points on the circumference of a circle, centre O. AC is a diameter of the circle. Angle DAC = 20º.

(a) Find the size of angle ACD.

(b) Find the size of angle DBC. Give a reason for your answer.

Question 6

Diagram NOT accurately drawn

A, B, C and D are points on the circle, centre O. Angle BOD = 86º

(b) Work out the size of angle BCD.

Question 7

A ,B, C and D are points on a circle, centre O. Angle BOD = 116°

(a) Calculate the size of angle BAD.

Given that BC = CD.

(b) Calculate the size of angle DBC.

Question 8

A, B, C and D are four points on the circumference of a circle. ABE and DCE are straight lines. Angle BAC = 25°. Angle EBC = 60°.

(a) Find the size of angle ADC.

(b) Find the size of angle ADB

Angle CAD = 65°. Ben says that BD is a diameter of the circle.