CBSE Class 9 Maths Lab Manual – Property of Cyclic Quadrilateral

CBSE Class 9 Maths Lab Manual – Property of Cyclic Quadrilateral

Objective
To verify that the opposite angles of a cyclic quadrilateral are supplementary by paper folding activity.

Prerequisite Knowledge

1. A quadrilateral whose all four vertices lies on the circle is known as cyclic quadrilateral.
2. Concept of opposite angles of a quadrilateral.
3. Concept of Supplementary angles.

Materials Required
White paper sheet, compass, glazed papers, pencil, a pair of scissors, gluestick.

Procedure

1. Draw a circle of radius 2 cm on a white glazed paper with centre O.
2. Cut this circle with centre O and draw one more circle of same radius.
3. Take any four points A, B, C, D on the circumference of both the circles.
4. Join AB, BC, CD, DA by paper folding on the both the circles.
5. We get a cyclic quadrilateral ABCD on both the circles [fig.(i) and fig.(ii)]. Take ∠A and ∠C of blue colour and ∠B and ∠D of pink colour in both circles.
   
   
6. From the second circle, using transparent sheets make cut outs of ∠A, ∠B, ∠C, ∠D. [fig. (iii)]. Take ∠A and ∠C of blue colour and ∠B and ∠D of pink colour.
   
7. Make a straight line on white paper sheet. Place cut outs of ∠A and ∠C adjacent to each other on a straight line and paste them [fig.(iv)].
8. Take cut outs of ∠B and ∠D and place them adjacent to each other on another straight line and paste them [fig.(iv)].
   

Observation
As ∠A and ∠C forms a linear pair.
∴ ∠A + ∠C = 180°
Similarly, ∠B + ∠D = 180°

Result
Hence, it is verified that in a cyclic quadrilateral, sum of opposite angles is 180°.

Learning Outcome
If a cyclic quadrilateral is a parallelogram then it become a rectangle, this can be proyed by paper folding and cutting method.

Activity Time
By same paper cutting activity students can verify that in a cyclic quadrilateral, the exterior angle is equal to the opposite interior angle.

Viva Voce

Question 1.
What are supplementary angles ?
Answer:
If the sum of two angles is 180°, then angles are supplementary angles.

Question 2.
What are complementary angles?
Answer:
If the sum of two angles is 90°, then angles are complementary angles.

Question 3.
What are adjacent angles of a quadrilateral ?
Answer:
Two angles lying on the same side of a quadrilateral.

Question 4.
What is the sum of four angles of a quadrilateral ?
Answer:
360°.

Question 5.
If one of the angles of a cyclic quadrilateral is 30°, then what will the value of its opposite angle ?
Answer:
150°

Question 6.
What do you mean by a cyclic quadrilateral ?
Answer:
A quadrilateral is called a cyclic if all the four vertices are concyclic.

Question 7.
If a cyclic quadrilateral is a parallelogram, then what will be the kind of parallelogram ?
Answer:
Rectangle.

Question 8.
What will the sum of opposite angles of a cyclic quadrilateral ?
Answer:
180°.

Question 9.
What will be the name of a quadrilateral if the pair of opposite angles is supplementary ?
Answer:
Cyclic quadrilateral.

Multiple Choice Questions

Question 1.
ABCD is a cyclic quadrilateral, in which ∠ABC = 90°. The value of ∠ADC is:
(i) 90°
(ii) 45°
(iii) 70°
(iv) none of these

Question 2.
In a circle with centre O, the angle subtended by arc BCD at the centre is 140°. BC is produced to P. Find ∠DCP:

(i) 70°
(ii) 35°
(iii) 280°
(iv) none of these

Question 3.
In the given figure, ∆ABC is an isosceles triangle with AB = AC and ∠ABC = 50°. Find ∠BDC:

(i) 100°
(ii) 25°
(iii) 80°
(iv) none of these

Question 4.
In the given figure, ∆DEF is an isosceles triangle with DE = DF and ∠DEF = 60°. Find ∠EAF:

(i) 120°
(ii) 30°
(iii) 300°
(iv) none of these

Question 5.
ABCD is a cyclic quadrilateral in which BC is parallel to AD, ∠ADC =110°, ∠BAC = 50°. Find ∠DAC
(i) 100°
(ii) 30°
(ii) 60°
(iv) none of these

Question 6.
ABCD is a cyclic quadrilateral. If ∠BCD = 100°, ∠ABD = 70°, the value of ∠ADB will be:
(i) 60°
(ii) 30°
(iii) 150°
(iv) none of these

Question 7.
If PQRS is a cyclic quadrilateral, ∠P = 3x°, ∠Q =y°, ∠R=x°, ∠S = 5y°, find the value of x° and y°.
(i) 45°, 30°
(ii) 90°, 60°
(iii) 90°, 30°
(iv) none of these

Question 8.
ABCD is a cyclic trapezium in which AD || BC, if ∠B = 70°, find the value of ∠A:
(i) 70°
(ii) 110°
(iii) 35°
(iv) none of these

Question 9.
If ABCD is a cyclic quadrilateral, in which ∠DBC = 70°, ∠BAC = 40°, find ∠BCD :
(i) 100°
(ii) 40°
(iii) 70°
(iv) none of these

Question 10.
Find the measure of the opposite angles of a cyclic quadrilateral if one of them is \(\frac { 11 }{ 4 }\) of the other:
(i) 48°, 132°
(ii) 29°, 132°
(iii) 48°,264°
(iv) none of these

Answers

1. (i)
2. (i)
3. (iii)
4. (i)
5. (ii)
6. (ii)
7. (i)
8. (ii)
9. (iii)
10. (i)

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