Lab Manual: Verify that the Opposite Angles of a Cyclic Quadrilateral - Class 9 PDF Download

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Objective

Materials Required
Cardboard, White paper, Drawing sheet, Geometry box, Scissors, Sketch pens, Adhesive, Transparent sheet

Prerequisite Knowledge
(i) Knowledge about the supplementary angles, linear pair axiom and interior opposite angles.
(ii) All the basic information related to cyclic quadrilateral.

Theory

• Two angles are said to be supplementary, if the sum of their measures is 180°. In the figure given below, ∠BOC and ∠AOC are supplementary angles, (see Fig. 25.1)
  
• If a ray stands on a line, then the sum of two adjacent angles so formed is 180°, i.e. the sum of the linear pair is 180°.
  In the above figure, ∠AOC + ∠BOC =180°.
  If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line or two opposite rays. The two axioms given above together are called the linear pair axiom.
• If the side BC of ΔABC is produced to D, then ∠AÇD is called an exterior angle of ΔABC at C, while ∠BAC and ∠ABC are called its interior opposite angles. It is denoted by exterior ∠ACD.
  
• A quadrilateral ABCD is called a cyclic quadrilateral, if all the four vertices A B, C and D are concyclic, i.e. A, B, C and D lie on a circle. In Fig. 25.4, ABCD is a cyclic quadrilateral. The sum of opposite angles of a cyclic quadrilateral is always 80°, i.e. they are supplementary.
  

Procedure

• Take a cardboard of suitable size and paste a white paper on it.
• Draw a circle of suitable radius on drawing sheet.
• Cut out the circle and paste it on cardboard.
• In the circle, draw a quadrilateral ABCD such that all the four vertices of quadrilateral lie on the circle. Name the angles as ∠A, ∠B, ∠C and ∠D. (see Fig. 25.5)
  
• Make the cut outs of the angles, (i.e. ∠A, ∠B, ∠C and ∠D) with the help of a transparent sheet, (see Fig. 25.6)

• By using adhesive, paste cut outs of the opposite angles ∠A and ∠C, ∠B and ∠D. (see Fig. 25.7)
  

Demonstration
Joining the opposite angles ∠A and  ∠C, ∠B and ∠D, we get straight angles, (see Fig. 25.4).

Hence, ∠A + ∠C = 180° and ∠B + ∠D = 180°.

Observation
By actual measurement,  ∠A = ……… , ∠B = ……… ,
∠C = ……… ,  ∠D = ……… ,
So, ∠A + ∠C = ……… , ∠B + ∠D = ……… .
Hence, sum of each pair of the opposite angles of a cyclic quadrilateral is ……… .

Result
We have verified that the opposite angles of a cyclic quadrilateral are always supplementary.

Application
This property can be used in solving various problems in geometry.