**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**

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

**Materials Required**

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

**Procedure**

- Draw a circle of radius 2 cm on a white glazed paper with centre O.
- Cut this circle with centre O and draw one more circle of same radius.
- Take any four points A, B, C, D on the circumference of both the circles.
- Join AB, BC, CD, DA by paper folding on the both the circles.
- 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.

- 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.

- 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)].
- 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**

- (i)
- (i)
- (iii)
- (i)
- (ii)
- (ii)
- (i)
- (ii)
- (iii)
- (i)

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