**NCERT Class 9 Maths Lab Manual – Verify that the Sum of the Angles of a Quadrilateral is 360°**

**OBJECTIVE**

To verify experimentally that the sum of the angles of a quadrilateral is 360°.

**Materials Required**

- Cardboard
- White paper
- Tracing paper
- Cutter/scissors
- Coloured drawing sheets
- Geometry box
- Adhesive
- Sketch pens

**Prerequisite Knowledge**

Concept of quadrilateral and its properties.

**Theory**

**Quadrilateral:**A closed figure having four sides, four angles and four vertices is called a quadrilateral.

Here, the term ‘Quad’ means ‘Four’ and term ‘Lateral’ means ‘Sides’, so that the term ‘Quadrilateral’ means ‘a figure bounded by four sides’.

In a quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices and ∠A, ∠B, ∠C and ∠D are the four angles formed at the vertices, (see Fig. 18.1).

**Terms Related to Quadrilateral****Opposite Sides:**Two sides of a quadrilateral which do not intersect, i.e. have no common end point (vertex) are called opposite sides. In quadrilateral ABCD, AB, CD and BC, AD are two pairs of opposite sides.**Consecutive or Adjacent Sides:**Two sides of a quadrilateral which have a common point, i.e. intersect each other are called consecutive sides. In quadrilateral ABCD, AB, BC; BC, CD;

CD, DA and DA, AB are four pairs of consecutive sides.**Opposite Angles:**Two angles of a quadrilateral are said to be opposite angles, if they do not have common arm. In quadrilateral ABCD, ∠A, ∠C and ∠B, ∠D are two pairs of opposite angles.**Consecutive or Adjacent Angles:**Two angles of a quadrilateral are said to be consecutive or adjacent angles, if they have a common arm. In quadrilateral ABCD, ∠A, ∠B; ∠B, ∠C; ∠C, ∠D and ∠D, ∠A are four pairs of consecutive angles.**Diagonal:**In a quadrilateral, the line segment joining the opposite vertices is called a diagonal of the quadrilateral. In quadrilateral ABCD, AC and BD are two diagonals.

- The sum of the four angles of a quadrilateral is 360°.

**Procedure**

- Take a piece of rectangular cardboard of suitable size and by using adhesive, paste a white paper on it.
- Cut out a quadrilateral from a drawing sheet and name it as ABCD. Now, using adhesive, paste it on cardboard, (see Fig. 18.2).

- Make cut outs of ∠A, ∠B, ∠C & ∠D of Quadrilateral ABCD with the help of tracing paper.(see in Fig.18.3).

- Arrange the four cut out angles at a point O. (see Fig.18.4).

**Demonstration**

- We came to know that the vertex of each cut out angle coincides at the point O.
- Such arrangement of cut outs indicates that the sum of the angles of a quadrilateral forms a complete angle, i.e. 360°.

**Observation**

Measures of ∠A = ………. ,

∠B = ………. ,

∠C = ………. ,

∠D = ………. ,

Sum of ∠A + ∠B + ∠C + ∠D = ………. .

**Result**

We have verified that the sum of the angles of a quadrilateral is a complete angle, i.e. 360°.

**Application**

This property may be useful in solving problems related to many types of quadrilaterals, such as parallelograms, trapeziums, rhombuses, squares and rectangles, etc.

**Viva Voce**

**Question 1:**

What is the angle sum property of a quadrilateral?

**Answer:**

The sum of all angles of a quadrilateral is a complete angle, i.e. 360°.

**Question 2:**

The sum of three angles of a quadrilateral is 280°. Find the measure of the fourth angle.

**Answer:**

Fourth angle = 360° – 280° = 80°

**Question 3:**

Is it true that every parallelogram is a rectangle?

**Answer:**

No, only those parallelogram is a rectangle whose all angles are 90°.

**Question 4:**

In which quadrilateral(s), diagonals are perpendicular to each other?

**Answer:**

Rhombus

**Question 5:**

Is it true that diagonals of a rhombus are equal?

**Answer:**

No

**Question 6:**

What are the conditions that any quadrilateral be a square?

**Answer:**

- All four sides of a quadrilateral are equal.
- Each angle of a quadrilateral is 90°.
- Diagonals are equal and bisect each other.

**Question 7:**

Is it true that parallelogram is always a trapezium but a trapezium is not always a parallelogram?

**Answer:**

True

**Question 8:**

How many vertices a quadrilateral has?

**Answer:**

A quadrilateral has 4 vertices.

**Question 9:**

Can all the angles of a quadrilateral be right angles? Give reason.

**Answer:**

Yes, all the angles of a quadrilateral can be right angles, e.g. Square and rectangle.

**Suggested Activity**

Verify experimentally the angle sum property for other types quadrilateral.

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