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## NCERT Class 8 Maths Textbook Chapter 3 With Answer Book PDF Free Download

### Chapter 3: Understanding Quadrilaterals

#### 3.1 Introduction

You know that the paper is a model for a plane surface. When you join a number of points without lifting a pencil from the paper (and without retracing any portion of the drawing other than single points), you get a plane curve. Try to recall different varieties of curves you have seen in the earlier classes.

#### 3.2.5 Angle sum property

Do you remember the angle-sum property of a triangle? The sum of the measures of the three angles of a triangle is 180°. Recall the methods by which we tried to visualize this fact. We now extend these ideas to a quadrilateral.

#### 3.4 Kinds of Quadrilaterals

Based on the nature of the sides or angles of a quadrilateral, it gets special names.

#### 3.4.6 Diagonals of a parallelogram

The diagonals of a parallelogram, in general, are not of equal length. (Did you check this in your earlier activity?) However, the diagonals of a parallelogram have an interesting property.

Author | NCERT |

Language | English |

No. of Pages | 20 |

PDF Size | 1722 KB |

Category | Mathematics |

Source/ Credits | ncert.nic.in |

### NCERT Solutions Class 8 Maths Chapter 3 Understanding Quadrilaterals

**1. Given here are some figures.**

**Classify each of them on the basis of the following.**

**Simple curve (b) Simple closed curve (c) Polygon**

**(d) Convex polygon (e) Concave polygon**

Solution:

a) Simple curve: 1, 2, 5, 6 and 7

b) Simple closed curve: 1, 2, 5, 6 and 7

c) Polygon: 1 and 2

d) Convex polygon: 2

e) Concave polygon: 1

**2. How many diagonals does each of the following have?**

**a) A convex quadrilateral (b) A regular hexagon (c) A triangle**

Solution:

a) A convex quadrilateral: 2.

b) A regular hexagon: 9.

c) A triangle: 0

**3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)**

Solution:

Let ABCD be a convex quadrilateral.

From the figure, we infer that the quadrilateral ABCD is formed by two triangles,

i.e. ΔADC and ΔABC.

Since, we know that sum of interior angles of triangle is 180°,

the sum of the measures of the angles is 180° + 180° = 360°

Let us take another quadrilateral ABCD which is not convex .

Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD. In ΔABC,

∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)

∴, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°

⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°

⇒ ∠A + ∠B + ∠C + ∠D = 360°

Thus, this property holds if the quadrilateral is not convex.

**4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)**

**What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 (d) n**

Solution:

The angle sum of a polygon having side n = (n-2)×180°

a) 7

Here, n = 7

Thus, angle sum = (7-2)×180° = 5×180° = 900°

b) 8

Here, n = 8

Thus, angle sum = (8-2)×180° = 6×180° = 1080°

c) 10

Here, n = 10

Thus, angle sum = (10-2)×180° = 8×180° = 1440°

d) n

Here, n = n

Thus, angle sum = (n-2)×180°

**5. What is a regular polygon?**

**State the name of a regular polygon of**

**(i) 3 sides (ii) 4 sides (iii) 6 sides **Solution:

Regular polygon: A polygon having sides of equal length and angles of equal measures is called a regular polygon. i.e., A regular polygon is both equilateral and equiangular.

(i) A regular polygon of 3 sides is called an equilateral triangle.

(ii) A regular polygon of 4 sides is called square.

(iii) A regular polygon of 6 sides is called a regular hexagon.

NCERT Class 8 Maths Textbook Chapter 3 With Answer Book PDF Free Download