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Angle Properties of Circles

Total questions: 40

Worksheet time: 34mins

Name
Class
Date
1.

O is the centre and PT is a tangent to the circle at T.

Find the value of x.

(a)  

2.

O is the centre and PT is a tangent to the circle at T.

Find the values of x and y.

a)

x = 45⁰, y = 90⁰

b)

x = 45⁰, y = 45⁰

c)

x = 90⁰, y = 45⁰

d)

x = 90⁰, y = 90⁰

3.

O is the centre and PT is a tangent to the circle at T.

Find the value of b.

(a)  

4.

O is the centre and PT is a tangent to the circle at T.

Find the values of a and b.

a)

a = 35⁰, b = 110⁰

b)

a = 55⁰, b = 70⁰

c)

a = 72.5⁰, b = 35⁰

d)

a = 35⁰, b = 55⁰

5.

O is the centre and PT is a tangent to the circle at T.

Find the values of p, q and r.

a)

p = 25⁰, q = 65⁰ and r = 25⁰

b)

p = 25⁰, q = 32.5⁰ and r = 57.5⁰

c)

p = 25⁰, q = 57.5⁰ and r = 32.5⁰

d)

p = 25⁰, q = 57.5⁰ and r = 32.5⁰

6.

O is the centre and PT is a tangent to the circle at T.

AOT and BOP are straight lines.

Find the values of a and b.

a)

a = 85⁰, b = 80⁰

b)

a = 80⁰, b = 80⁰

c)

a = 85⁰, b = 85⁰

d)

a = 80⁰, b = 85⁰

7.

O is the centre, PA is a tangent to the circle at A.

AOC and PBC are straight lines.

Find the values of x, y and z.

a)

x = 60⁰, y = 30⁰

and z = 90⁰

b)

x = 30⁰, y = 60⁰

and z = 90⁰

c)

x = 60⁰, y = 60⁰

and z = 90⁰

d)

x = 90⁰, y = 30⁰

and z = 60⁰

8.

O is the centre, PA is a tangent to the circle at A.

EOC and DOA are straight lines.

Find the values of x, y and z.

a)

x = 100⁰, y = 40⁰

and z = 50⁰

b)

x = 140⁰, y = 20⁰

and z = 70⁰

c)

x = 120⁰, y = 30⁰

and z = 60⁰

d)

x = 150⁰, y = 15⁰

and z = 75⁰

9.

O is the centre, PA and QA are tangents to the circle at P and Q respectively.

Find the values of a, b and c.

a)

a = 75⁰, b = 75⁰

and c = 60⁰

b)

a = 80⁰, b = 80⁰

and c = 50⁰

c)

a = 45⁰, b = 45⁰

and c = 30⁰

d)

a = 90⁰, b = 90⁰

and c = 30⁰

10.

O is the centre, PA and QA are tangents to the circle at P and Q respectively.

Find the values of a, b and c.

a)

a = 11.5⁰, b = 75⁰

and c = 37.5⁰

b)

a = 23⁰, b = 90⁰

and c = 67⁰

c)

a = 23⁰, b = 90⁰

and c = 23⁰

d)

a = 11.5⁰, b = 60⁰

and c = 30⁰

11.

O is the centre of the circle.

PS and PT are tangents to the circle at S and T respectively.

Find the value of b.

(a)  

12.

O is the centre of the circle.

PS and PT are tangents to the circle at S and T respectively.

Find the value of c.

(a)  

13.

O is the centre of the circle.

PS and PT are tangents to the circle at S and T respectively.

Find the perimeter of quadrilateral OTPS.

(a)  

14.

O is the centre of the circle.

PS and PT are tangents to the circle at S and T respectively.

Find the area of quadrilateral OTPS.

(a)  

15.

O is the centre of the circle.

PA and PB are tangents to the circle at A and B respectively.

Find the value of x.

(a)  

16.

O is the centre of the circle.

PA and PB are tangents to the circle at A and B respectively.

Find the value of x.

(a)  

17.

O is the centre of the circle.

PA and PB are tangents to the circle at A and B respectively.

AODC and PBC are straight lines.

Find the value of a, b and c.

a)

a = 55⁰, b = 70⁰ 

and c = 90⁰

b)

a = 55⁰, b = 70⁰ 

and c = 55⁰

c)

a = 65⁰, b = 35⁰ 

and c = 90⁰

d)

a = 35⁰, b = 35⁰ 

and c = 55⁰

18.

Given that BD = DC, ∠ABD = 42⁰ and ∠DBC = 65⁰,

calculate

∠BDC

(a)  

19.

Given that BD = DC, ∠ABD = 42⁰ and ∠DBC = 65⁰,

calculate

∠BAC

(a)  

20.

Given that BD = DC, ∠ABD = 42⁰ and ∠DBC = 65⁰,

calculate

∠DAB

(a)  

21.

Given that BD = DC, ∠ABD = 42⁰ and ∠DBC = 65⁰,

calculate

∠DEC

(a)  

22.

Find the values of x and y.

a)

x = 30⁰, y = 30⁰

b)

x = 80⁰, y = 30⁰

c)

x = 30⁰, y = 80⁰

d)

x = 15⁰, y = 40⁰

23.

Find the values of c and d.

a)

c = 30⁰, d = 30⁰

b)

c = 30⁰, d = 65⁰

c)

c = 65⁰, d = 65⁰

d)

c = 65⁰, d = 30⁰

24.

Find ∠ACB

(a)  

25.

Find ∠ADB

(a)  

26.

Find ∠ACD

(a)  

27.

Find ∠BAE

(a)  

28.

Find ∠BDC

(a)  

29.

Find the values of x and y.

a)

x = 30⁰, y = 36⁰

b)

x = 36⁰, y = 36⁰

c)

x = 24⁰, y = 24⁰

d)

x = 36⁰, y = 24⁰

30.

Find the value of x

(a)  

31.

Find the value of y.

(a)  

32.

Find the value of a.

(a)  

33.

Given that O is the centre of the circle and

∠OBC = 20⁰,

calculate ∠BAC.

(a)  

34.

Find ∠ACB

(a)  

35.

In the diagram, O is the centre of the circle, and SP is parallel to RQ.

Calculate ∠PSR

(a)  

36.

In the diagram, O is the centre of the circle, and SP is parallel to RQ.

Calculate ∠OPQ

(a)  

37.

O is the centre of the circle and ABE is a triangle.

If the circle cuts BE at C and AE at D,

calculate ∠x, ∠y and ∠z.

a)

x = 25⁰, y = 140⁰

and z = 15⁰

b)

x = 30⁰, y = 110⁰

and z = 40⁰

c)

x = 40⁰, y = 110⁰

and z = 30⁰

d)

x = 15⁰, y = 140⁰

and z = 25⁰

38.

In the diagram, P, Q, R and S are points on the circle.

PS and QR are produced to meet at A.

If ∠QSR = 35⁰, ∠QPA = 70⁰ and ∠RSA = 75⁰,

calculate ∠PQR.

(a)  

39.

In the diagram, P, Q, R and S are points on the circle.

PS and QR are produced to meet at A.

If ∠QSR = 35⁰, ∠QPA = 70⁰ and ∠RSA = 75⁰,

calculate ∠PAQ.

(a)  

40.

Given that OP = OQ, find the value of a.

(a)