The experiment to determine Verify that the Opposite Angles of a Cyclic Quadrilateral are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.
Maths Lab Manual Class 9 CBSE Verify that the Opposite Angles of a Cyclic Quadrilateral Experiment
Determine Verify that the Opposite Angles of a Cyclic Quadrilateral Class 9 Practical
OBJECTIVE
To verify that the opposite angles of a cyclic quadrilateral are supplementary.
Materials Required
- Cardboard
- White paper
- Drawing sheet
- Geometry box
- Scissors
- Sketch pens
- Adhesive
- Transparent sheet
Prerequisite Knowledge
- Knowledge about the supplementary angles, linear pair axiom and interior opposite angles.
- 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.
Viva Voce
Question 1:
What do you understand by the term a cyclic quadrilateral?
Answer:
A quadrilateral having all the vertices on the boundary of the circle is called a cyclic quadrilateral.
Question 2:
What is the sum of each pair of opposite angles of a cyclic quadrilateral?
Answer:
180°
Question 3:
If one of the angles of a cyclic quadrilateral is 40°, then what will be the value of its opposite angle?
Answer:
140°
Question 4:
If a cyclic quadrilateral is a parallelogram, then what is the type of parallelogram?
Answer:
Rectangle
Question 5:
Is the sum of adjacent angles of a cyclic quadrilateral 180°?
Answer:
No, only the sum of opposite angles of a cyclic quadrilateral is 180°.
Question 6:
What is the name of quadrilateral, if each pair of opposite angles is supplementary?
Answer:
Cyclic quadrilateral
Question 7:
What is the type of quadrilateral formed by the internal angle bisectors of cyclic .quadrilateral?
Answer:
Cyclic quadrilateral
Question 8:
Which property has to be added in a trapezium for making it a cyclic quadrilateral?
Answer:
Non-parallel sides of a trapezium should be equal.
Suggested Activity
Verify that the exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
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