# Angle Properties of a Circle - Mathematics Form 2 Notes

## 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

• 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 = 50o and angle ACD = 20o.

Calculate;
1. ∠OAB

Solution

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

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
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

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

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