Angle Properties of a Circle - Mathematics Form 2 Notes

Share via Whatsapp


Arc, Chord and Segment of a Circle


  • Any part on the circumference of a circle is called an arc. We have the major arc and the minor Arc as shown below.
    major and minor arc


  • A line joining any two points on the circumference.
  • A chord divides a circle into two regions called segments, the larger one is called the major segment the smaller part is called the minor segment.
    major segment

Angle at the Centre and Angle on the Circumference

  • The angle which the chord subtends to the centre is twice that it subtends at any point on the circumference of the circle.
    angle at centre

Angle in the Same Segments

  • Angles subtended on the circumference by the same arc in the same segment are equal.
  • Also note that equal arcs subtend equal angles on the circumference
    angle in same segments

Cyclic Quadrilaterals

  • Quadrilateral with all the vertices lying on the circumference are called cyclic quadrilateral

Angle Properties of Cyclic Quadrilateral

  • The opposite angles of cyclic quadrilateral are supplementary hence they add up to 1800.
  • If a side of quadrilateral is produced the interior angle is equal to the opposite exterior angle.
    cyclic quadrilateral


In the figure below ∠ ADE = 1200 find ∠ ABC
cyclic quadrilateral example


Using this rule, If a side of quadrilateral is produced the interior angle is equal to the opposite exterior angle.
Find ∠
ABC = 1200
Angles formed by the diameter to the circumference is always 900

cyclic quadrilateral solution


  • Angle in semicircle = right angle
  • Angle at centre is twice than at circumference
  • Angles in same segment are equal
  • Angles in opposite segments are supplementary


  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.
    angle properties example
    1. ∠OAB
    2. ∠ADC


  1. AOB = 2ACB
    = 100o
    OAB = 180 – 100 Base angles of Isosceles ∆
    = 40
  2. BAD = 1800 700
    = 110o

Past KCSE Questions on the Topic.

  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.
    angle properties q1
    Giving reasons for your answer, find the size of :-
    1. Angle ACB
    2. Angle AOD
    3. Angle CAB
    4. Angle ABC
    5. Angle AXB
  2. In the figure below CP= CQ and ∠CQP = 1600. If ABCD is a cyclic quadrilateral, find ∠BAD
    angle properties q2
  3. In the figure below AOC is a diameter of the circle centre O; AB = BC and ∠ ACD = 250, EBF is a tangent to the circle at B.G is a point on the minor arc CD.
    angle properties q3
    1. Calculate the size of
      1. ∠ BAD
      2. The Obtuse ∠ BOD
      3. ∠ BGD
    2. Show the ∠ ABE = ∠CBF. Give reasons
  4. In the figure below PQR is the tangent to circle at Q. TS is a diameter and TSR and QUV are straight lines. QS is parallel to TV. Angles SQR = 40o and angle TQV = 55o
    angle properties q4
    Find the following angles, giving reasons for each answer
    1. QST
    2. QRS
    3. QVT
    4. UTV
  5. In the figure below, QOT is a diameter. QTR = 480, TQR = 760 and SRT = 370
    angle propertiesq5
    1. ∠RST
    2. ∠SUT
    3. Obtuse ∠ROT
  6. In the figure below, points O and P are centers of intersecting circles ABD and BCD respectively. Line ABE is a tangent to circle BCD at B. Angle BCD = 420
    angle properties q6
    1. Stating reasons, determine the size of
      1. ∠CBD
      2. Reflex ∠BOD
    2. Show that ∆ ABD is isosceles
  7. The diagram below shows a circle ABCDE. The line FEG is a tangent to the circle at point E. Line DE is parallel to CG, ∠ DEC = 280 and ∠ AGE = 320
    angle properties q7
    1. ∠ AEG
    2. ∠ ABC
  8. In the figure below R, T and S are points on a circle centre OPQ is a tangent to the circle at T. POR is a straight line and ∠QPR = 200
    angle properties q8
    Find the size of ∠RST
Join our whatsapp group for latest updates

Download Angle Properties of a Circle - Mathematics Form 2 Notes.

Tap Here to Download for 50/-

Why download?

  • ✔ To read offline at any time.
  • ✔ To Print at your convenience
  • ✔ Share Easily with Friends / Students

Get on WhatsApp Download as PDF
Subscribe now

access all the content at an affordable rate
Buy any individual paper or notes as a pdf via MPESA
and get it sent to you via WhatsApp


What does our community say about us?

Join our community on:

  • easyelimu app
  • Telegram
  • facebook page
  • twitter page
  • Pinterest