Angles

92.
The diagram shows a circle centre O and radius 5 cm.
AB is a tangent and AB=12 cm.
NOT TO SCALE O 5 cm A 12 cm B

Find OB.
................................. [2]
Paper 1
93.
The diagram shows points B, D and E lying on a circle.
AC is a tangent at B and angle EAB=55°.
AD is a straight line and BD is a diameter.
NOT TO SCALE A B C D E 55°

(a) State the reason why angle DEB=90°.
(b) Calculate the size of angle EBD.
(a) ......................... [1] (b) ......................... [2]
Paper 1
94.
In the diagram, PQ and RS are parallel lines.
NOT TO SCALE P Q R S 70° b 30° a

Calculate the angles a and b.
a= ................................. b= ................................. [3]
Paper 1
95.
Points A, B and C lie on the circumference of a circle, centre O.
BOC=90° and AB=10 cm.
NOT TO SCALE A B C

(a) State with a reason the value of angle OBC.
(b) Calculate AC, giving your answer correct to the nearest integer.
(c) State the special name of the triangle ABC.
(a) ......................... [1] (b) ......................... [1] (c) ......................... [3]
Paper 1
96.
The diagram shows figure ABCDE.
AC=BC
AE^B=37°
AC^B=50°
AE is parallel to BC
NOT TO SCALE 37° 50° A B C D E

(a) Find angle CAE and give the reason for your answer.
(b) Find angle ADE and give the reason for your answer.
(a) ......................... [2] (b) ......................... [2]
Paper 1
97.
The diagram shows points A, B, C, and D on the circumference of a circle centre O.
Angle BOD=124°.
NOT TO SCALE 124° A B C D O

Write the size of:
(a) Angle DCB
(b) Angle DAB
(a) ......................... [1] (b) ......................... [1]
Paper 1
99.
In the diagram, HMT is a straight line. NT is a tangent to the circle at N and HN is a diameter to the circle.
39° H M N T

(i) Give a reason why angle HMN=90°
(ii) Find the size of angle HTN.
(i) ......................... [1] (ii) ......................... [2]
Paper 1
101.
In the triangle ABC, AB is parallel to DE.
Angle BAD=110° and angle ACB=36°.
NOT TO SCALE 110° 36° A B C D E

Find the size of the angle:
(a) ABC
(b) CDE
(c) DEB
(a) ......................... [1] (b) ......................... [1] (c) ......................... [1]
Paper 1
103.
The diagram shows a circle with points A, B and T on the circumference.
BT is the diameter of the circle, TAC is a straight line and BC is a tangent.
Angle ABT=60°.
NOT TO SCALE 60° A B C T

Find:
(a) angle BTA
(b) angle BCA
(a) ......................... (b) .........................
Paper 1
107.
The diagram shows two regular polygons joined at side XY. AD is a straight line.
NOT TO SCALE 6 3 4 P Q R S T

Calculate:
(a) The size of angle XBC
(b) The size of angle ECD
(a) ......................... (b) .........................
Paper 1
53.
The diagram shows a triangle OPQ and circle, centre O with radius r cm.
The circle cuts OP and OQ at A and B respectively.
AP = 7 cm, PQ = 5 cm, QB = 6 cm and angle OQP = 90°.

NOT TO SCALE O A B P Q r cm 7 cm r cm 6 cm 5 cm

(a) Write an expression, in terms of r, for
sin ⁡ ∠ P O Q .
......................... [1]
(b) Show that r = 6 cm.
......................... [3]
(c) Calculate:
(i) The length of major arc AB,
......................... cm [2]
(ii) The shaded area.
......................... cm2 [2]
Paper 3
54.
The diagram shows a circle, centre O, radius 5 cm. RTP is a tangent to the circle at T, and TP = 12 cm.
R T P O 30° 5 12

(a) Calculate:
(i) OP,
......................... cm [2]
(ii) angle OPT,
......................... [2]
(iii) angle ROP.
......................... [2]
(b) The diagram shows a circle, centre O, radius 8 cm.
Angle MON = 108°.
NOT TO SCALE M N O 108° 8

Calculate:
(i) The area of the minor sector MON,
......................... cm2 [2]
(ii) The perimeter of the major sector MON.
......................... cm [2]
Paper 3
55.
In the diagram, AD is parallel to EH, and BF is parallel to CG.
Angle BFE = 123° and angle BGF = 57°.
A B C D E F G H 123° 57°

(a) Find the size of angle BFG.
......................... [1]
(b) Calculate the size of angle BGC and give a reason for your answer.
......................... Reason: ......................... [2]
(c) Calculate the size of angle BCG and hence, write a special name for triangle BGC.
......................... Name: ......................... [2]
Paper 3