Angle Properties of Circles

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  • Last Update June 19, 2021


  1. Angle Properties of Circles

  1. In the figure below PQR and S are points on the circumference of a circle centre O. The point TSO and Q lie on a straight line MPT is a tangent to the circle at P.

Find the values of the following angles stating reasons in each case.

(a) ÐSRP                                                                                                                                 (2mks)

(b) ÐORP                                                                                                                                (2mks)

(c) ÐRPT                                                                                                                                 (2mks)

(d) ÐSTP                                                                                                                                 (2mks)

(e) ÐQPM                                                                                                                               (2mks)

  1. In the figure below, TA is a tangent to the circle ABCD with centre O. TAD = 480 and BOD = 1160

Giving reasons calculate:

  1. ACD (2mks)
  2. ABO (2mks)
  3. ADO (2mks)
  4. ACB (2mks)
  5. ATB (2mks)
  6. In the figure below AB = 8cm and O is the centre of the circle. Determine the area of the circle if ÐOAB = 15o (3mks)

The figure above is a cyclic quadrilateral PQRS. Given that TPX is a tangent at P and O is the centre of the circle and that RQX is a straight line with ÐRPQ = 50o and ÐPRS = 25o, giving reason in each case find:

(a)        angle PRQ                                                                                                                   (2mks)

(b)        angle PSR                                                                                                                   (2mks)

(c)        angle PXQ                                                                                                                  (2mks)

(d)       angle TPS                                                                                                                    (2mks)

(e)        angle POS                                                                                                                   (2mks)

  1. In the figure below ABCD is a circle with centre O. AB and DC meet at a point E outside the circle. DC = BC and








Find the angles

  1. BAD          (1mk)
  2. BDC          (1mk)
  3. BEC          (1mk)
  4. In the figure O and P are centres of intersecting circles ABD and DBC respectively. Line ABE is a tangent to circle BCD at B and angle BCD = 420.

Giving reasons determine the size of:

(a) Angle CBD.                                                                                                                       (2mks)

(b) Angle ODB.                                                                                                                      (2mks)

(c) Angle BAD.                                                                                                                      (2mks)

(d) Angle ABD                                                                                                                       (2mks)

(e) Angle ODA.                                                                                                                      (2mks)

  1. In the figure below, o is the centre of the circle. Express the angle w in terms of angles p and q.                                                                                                                                                             (2mks)
  2. Two circles of radii 4cm and 6cm intersect as shown below. If angle XBY = 30o and

angle XAY = 97.2o.

Find the area of the shaded part.

(Take = p 22  )


  1. In the diagram, O is the centre of the circle and AD is parallel to BC. If angle ACB =50o

and angle    ACD  = 20o.







Calculate; (i) ÐOAB

(ii) ÐADC

  1. Two intersecting circles have centres S and R. Given that their two radii are 28cm and 35cm,

their common chord AB = 38cm and angles ASB = 85.46° and ARB = 65.76°,

Calculate the shaded area

  1. In the figure below ABCD is a cyclic quadrilateral in which AD = DC and AB is parallel

to CD. Given that angle ABC = 80°, Find the size of:


  1. a) ÐDAC





  1. b) ÐBAC



  1. c) ÐBCD


  1. Line QR = 6.5cm is given below:-(Do not use a protractor for this question)

(a) Draw triangle PQR such that p lies above line QR, Ð PQR = 30o and PQ = 7cm

(b) By accurate construction on the diagram above, show the locus of a point which lies

within the  triangle such that:-

(i) T is more than 2.5cm from line PQ


(ii) T is not more than 4.5cm from Q

Shade the region in which T lies


(c) Lines QP and QR are produced to K and M respectively

(i) Show by construction on the diagram above, the locus of a point C which is

equidistant from  each of the lines PK, PR and RM

(ii) With centre C and an appropriate radius, draw a circle to touch each of the lines

PK, PR and RM  only once

Measure the radius

What name is given to the circle drawn in (c) (ii) with respect to triangle QPR

  1. The figure below shows a circle centre O and a cyclic quadrilateral ABCD. AC = CD, angle

ACD is 80o and BOD is a straight line. Giving reasons for your answer, find the size of :-









(ii) Angle AOD                                                                                                                                 (i) Angle ACB

(iii) Angle CAB

(iv) Angle ABC

(v) Angle AXB

  1. The figure below shows two circles of equal radius of 9 cm with centres A and B.

Angle CAD = 80o


  1. a) Calculate the area of:-
  2. i) The sector CAD. ii) The triangle CAD.                                                                                                  iii) The shaded region.


  1. In the diagram below, ÐQOT is a diameter. ÐQTP = 48o, ÐTQR = 46o and ÐSRT = 37o













Calculate, giving reasons in each case:-

(a) ÐRST

(b) ÐSUT

(c) ÐROT

(d) ÐPST

(e) Reflex ÐSOP


  1. The diagram below shows a circle with a chord PQ= 3.4cm and angle PRQ=40o.

Calculate the area of the shaded segment.













  1. The figure below shows circle ABCD. The line EDF is a tangent to the circle at D.

ÐADF = 70o ÐFAD = 65o and ÐCDE = 35o















Find the values of the following angles, stating your reasons in each case

(a) ÐABC

(b) ÐBCD

(c) ÐDCE

(d) ÐACD


  1. In the figure below BD is the diameter of the circle and O is the centre.





Find the size of


(a) ∠ADC

(b)  ∠ AEB



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