**NCERT Class 9 Maths Lab Manual – Verify that the Opposite Angles of a Cyclic Quadrilateral**

**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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