Angle Properties of a Circle - Mathematics Form 2 Notes

• Introduction
  • Arc, Chord and Segment of a Circle
• Angle at the Centre and Angle on the Circumference
• Angle in the Same Segments
• Cyclic Quadrilaterals
  • Angle Properties of Cyclic Quadrilateral
• Past KCSE Questions on the Topic.

Introduction

Arc, Chord and Segment of a Circle

Arc

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

Chord

• 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.
  
  

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 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
  
  

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.
  

Example

In the figure below ∠ ADE = 120⁰ find ∠ ABC

Solution

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

Summary

• 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

Example

1. In the diagram, O is the centre of the circle and AD is parallel to BC. If angle ACB = 50^o and angle ACD = 20^o.
   
   Calculate;
   1. ∠OAB
   2. ∠ADC

Solution

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

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 80^o and BOD is a straight line.
   
   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
   
   
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.
   
   
   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 = 40^o and angle TQV = 55^o
   
   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
   
   Calculate
   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
   
   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
   
   Calculate
   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. 
   
   Find the size of ∠RST