Lesson Notes By Weeks and Term - Senior Secondary School 2

THEOREMS AND PROOF RELATING TO CYCLIC QUADRILATERAL 

SUBJECT: MATHEMATICS

CLASS:  SS 2

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

  • New General Mathematics SSS2 by M.F. Macraeetal.
  • Essential Mathematics SSS2 by A.J.S. Oluwasanmi. 

 

 
WEEK NINE

TOPIC: THEOREMS AND PROOF RELATING TO CYCLIC QUADRILATERAL 

CONTENT

    -Definition of Cyclic Quadrilateral 

    -Theorems and proof relating to cyclic quadrilateral

    -Corrolary from Cyclic Quadrilateral

    -Solving problems on Cyclic Quadrilateral

 

CYCLIC QUADRILATERAL

Definition: A cyclic quadrilateral is described as any quadrilateral having its vertices lying on certain parts of the circumferences of a circle. i..e its four vertices. 









Note: that opposite angles of a cyclic quadrilateral lies in opposite segment of a circle.

 

Theorem:

The opposite angle of a cyclic quadrilateral are supplementary “or angle in opposite segment are supplementary i.e. They sum up to 1800.

 

Proof:

Given: A cyclic quadrilateral ABCD.

To prove:< BAD +< BCD = 1800







Construction: join B and D to O the centre

Proof:< BOD = 2y (angle of centre = 2 x angle at circumference)

Reflex< BOD = 2x (angle at centre = 2x angle at circumference)

2x + 2y = 3600 (angle at a point)

2(x + y) = 3600

x + y =  3600

                2

x + y = 1800

0

 

Example:Find the value of x






   x + 720=1800   (opp. Angle of a cyclic quadrilateral)

   x = 1800-720 = 1080

 

Evaluation

Find x and y

  1. 2.








Corollary from Cyclic Quadrilateral

Theorem:

The exterior angle of a cyclic quadrilateral to the interior opposite angle.

 

Proof:

Given: A cyclic quadrilateral ABCD 

To Prove: x1 = x2 or x2 = x1

 

Construction: Extend DC to x

Proof:  x1 + y = 1800 (opp. Angle in a cyclic quad)

    x2 + y = 1800 (angle in a straight line)

    x1 = x2 = (180-y)

    < BCX =< BAD    

 

Example:

In the fig. below PQRS are points on a circle centre O. QP is produced to x if< XPS = 770 and












0 (ext angle of a cyclic quadrilateral)

< QPS = 1800 – 770 = 1030 (angle on a straight line)

0(the exterior angle of a cyclic quad=interior opp.angle)

0=1540(angle at the centre=2×angle at the circumference)

0 – (1540 + 1030 + 680)  sum of angle in a quadrilateral 

PQO  = 3600 – 3250

PQO = 350

 

Example 








BEC is a triangle

BCE = 1800 – 850 (angle on a straight line)

CBE = 620 (exterior angle of cyclic quadrilateral)

 x =< BEC = 1800 – (620 + 950) [sum of angles in a ∆ ]

1800 – 1570 = 230

 

Evaluation

In the figure below AB is a diameter of semi circle ABCD. If 0,calculate









2.

   




In the fig, A,B,C,D are points on a circle such that 0.CD is produced to E  so that 0.Calculate

   

 

Application of Cyclic Quadrilateral [Circle Geometry]








Solution

0 (base angle of Isosceles  triangle ONM)

< NOM = 180 – (20 + 20)[ sum of angle in a triangle 1800 – 400 = 1400]

1400= 700 (2x angle at circum = angle at centre)

        2 

0 (base angle of  Isos triangle MNT)

0 (fe. 32 + 32) (extension of triangle MNT)

0 + 640) sum of angle in a triangle 

 

Evaluation

Find the marked angle.

   








GENERAL EVALUATION/REVISION QUESTIONS

Find the marked angle in each of the following.Where a point O is the centre of the circle.                                                               

  1.                                                                                           2.                             

 

   





  1. A right pyramid on a base 8cm square has a slant edge of 6cm, calculate the volume of the pyramid.
  2. Calculate the volume and total surface area of a cylinder which has a radius of 12cm and height 6cm

 

READING ASSIGNMENT

Essential Mathematics SSS2, pages 143-144, Exercise10.5,numbers 6-10.

 

WEEKEND  ASSIGNMENT

Objective

1.In the diagram below, O is the centre of the circle,

Calculate

(a) 1000    (b) 860    (c) 940    (d) 1440









2.In the diagram |PS| is a diameter of circle PQRS.  |PQ| =|QR|and0 find

(a) 320    (b) 370     (c) 480     (d) 530








3.In the diagram below, O is the centre of the Circle PQRS and

(a) 360 (b) 1440 (c) 720 (d) 1080







4.In the diagram below: PQRS is a cyclic quadrilateral, 0 and 0, Calculate

(a) 430 (b) 480 (c) 530 (d)580

   

   




5.In the diagram below; 0 is the centre of the circle. If

(a) 1050 (b) 750 (c) 150 (d) 1500









Theory

1.In the fig.Calculate the value of x giving a reason for each step in your answer.











  1. I n the diagram below, 0.Find






© Lesson Notes All Rights Reserved 2023