# Class 8th maths Chapter 3 Understanding of Quadrilaterals | 8th class mathematics

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**NCERT SOLUTIONS FOR CLASS 8 MATHS CHAPTER 3**

Understanding of Quadrilaterals

8th class mathematics class 8th maths solution

Understanding of Quadrilaterals

8th class mathematics class 8th maths solution

## NCERT SOLUTIONS FOR CLASS 8 MATHS CHAPTER 3

Understanding of Quadrilaterals

8th class mathematics Ex 3.1

1. Given here are some figures.

Classify each of them on the basis of the following.

(a) Simple curve | (b) Simple closed curve | (c) Polygon |

(d) Convex polygon | (e) Concave polygon |

Solution:

(a) Simple curve – 1, 2, 5, 7

(b) Simple closed curve – 1, 2, 5, 6, 7

(c) Polygon – 1, 2

(d) Convex polygon – 2

(e) Concave polygon – 1, 4

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: The sum of the measures of the angles of a convex quadrilateral is 360°.

The above quadrilateral is not convex. This quadrilateral has been divided into two triangles. The value of each triangle will be 180°. Therefore, the sum of the angles of this quadrilateral will also be 360°.

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

Figure | ||||

Side | 3 | 4 | 5 | 6 |

Angle sum | 180º | 2 × 180°= (4 – 2) × 180° | 3 × 180°= (5 – 2) × 180° | 4 × 180°= (6 – 2) × 180° |

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:

(a) Number of sides (x) = 7

The sum of all the interior angles of a polygon can be found by the following formula = (x – 2) × 180°

Put the value of the x

= (7 – 2) × 180

= 5 × 180

= 900°

(b) Number of sides (x) = 8

Sum of all interior angles of polygon = (x – 2) × 180°

Put the value of the x

= (8 – 2) × 180

= 6 × 180

= 1080°

(c) Number of sides (x) = 10

Sum of all interior angles of polygon = (x – 2) × 180°

Put the value of the x

= (10 – 2) × 180

= 8 × 180

= 1440°

(d) Number of sides (x) = n

Sum of all interior angles of polygon = (x – 2) × 180°

Put the value of the x

= (n – 2) × 180

5. What is a regular polygon?

State the name of a regular polygon of

3 sides | 4 sides | 6 sides |

Solution:

Regular Polygon : A polygon whose all sides and interior angles are equal is called a regular polygon.

3 Sides : A polygon whose all three sides are equal is called a triangle (equilateral triangle).

4 Sides : A polygon whose all four sides are equal is called a quadrilateral (square).

6 Sides : A polygon whose all six sides are equal is called a regular hexagon.

6. Find the angle measure x in the following figures.

(a)

Solution:

∵ Sum of four angles of quadrilateral = 360°

Hence 50° + 130° + 120° + x = 360°

300° + x = 360°

x = 360° – 300°

x = 60°

(b)

Solution:

y + z = 180° (linear angle pair)

90° + z = 180°

z = 180° – 90°

z = 90°

∵ Sum of four angles of quadrilateral = 360°

70° + 60° + 90° + x = 360°

220° + x = 360°

x = 360° – 220°

x = 140°

Solution:

70° + y = 180° (linear angle pair)

y = 180° – 70°

y = 110°

60° + z = 180° (linear angle pair)

z = 180° – 60°

z = 120°

∵ sum of all interior angles of a pentagon = (n − 2) × 180°

n = 5

= (5 – 2) × 180°

= 3 × 180°

= 540°

therefore, 110° + 120° + x + x + 30° = 540°

260° + 2x = 540°

2x = 540° – 260°

2x = 280°

x = 280° ÷ 2

x = 140°

(d)

Solution:

∵ sum of the five angles of a pentagon = 540°

therefore, x + x + x + x + x = 540°

5x = 540°

x = 540° ÷ 5

x = 108°

7. (a)

Solution:

z + 30° = 180° (linear angle pair)

z = 180° – 30°

z = 150°

x + 90° = 180° (linear angle pair)

x = 180° – 90°

x = 90°

The sum of the three exterior angles of a triangle = 360°

x + 120° = 180° (linear angle pair)

x = 180° – 120°

x = 60°

y + 80° = 180° (linear angle pair)

y = 180° – 80°

y = 100°

z + 60° = 180° (linear angle pair)

z = 180° – 60°

z = 120°

∵ Sum of four angles of quadrilateral = 360°

x + y + z + w = 360°

60° + 100° + 120° + w = 360°

280° + w = 360°

w = 360° – 280°

w = 80°

So, x + y + z + w = 360°

EXERCISE 3.2

1. Find x in the following figures.

(a)

Solution:

∵ Sum of exterior angles of any polygon = 360°

therefore, 125° + 125° + x = 360°

250° + x = 360°

x = 360° – 250°

x = 110°

(b) Solution:

∵ Sum of exterior angles of pentagon = 360°

Therefore, 70° + x + 90° + 60° + 90° = 360°

310° + x= 360°

x = 360° – 310°

x = 50°

2. Find the measure of each exterior angle of a regular polygon of

(i) 9 sides | (ii) 15 sides |

Solution:

(i) 9 sides

measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{n}\)
n = 9

Therefore, the measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{9}\)
= 40°

(ii) 15 sides

measure of each exterior angle of a regular polygon = = \(\displaystyle \frac{{360}}{n}\)
n = 15

Therefore, the measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{15}\)
= 24°

3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?

Solution:

measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{n}\)
24 = \(\displaystyle \frac{{360}}{n}\)
24n = 360°

n = \(\displaystyle \frac{{360}}{24}\)
n = 15

4. How many sides does a regular polygon have if each of its interior angles is 165°?

Solution:

each interior angle of a regular polygon = \(\displaystyle \frac{{\left( {n-2} \right)180}}{n}\)
165n = (n – 2) × 180°

165n = 180n – 360

165n – 180n = – 360

– 15n = – 360

n = \(\displaystyle \frac{{360}}{{15}}\)
n = 24

5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

(b) Can it be an interior angle of a regular polygon? Why?

Solution:

(a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?

measure of each exterior angle of a regular polygon = \(\displaystyle \frac{{360}}{n}\)
22 = \(\displaystyle \frac{{360}}{n}\)
22n = 360

n = \(\displaystyle \frac{{360}}{22}\)
n = 16.37

Therefore, it is not possible to have a polygon whose each exterior angle is 22° because here the values are completely divisible.

(b) Can it be an interior angle of a regular polygon? Why?

each interior angle of a regular polygon = \(\displaystyle \frac{{\left( {n-2} \right)180}}{n}\)
22n = (n – 2) × 180

22n = 180n – 360

22n – 180n = – 360

– 158n = – 360

n = 360 ÷ 158

n = 2.27

Therefore, it is not possible to have a polygon whose each interior angle is 22° because here the values are completely divisible.

6. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

Solution:

(a) What is the minimum interior angle possible for a regular polygon? Why?

We know that a regular polygon is made up of at least 3 sides. Therefore, it will be an equilateral triangle, each angle of which will be 60°.

x + x + x = 180°

3x = 180

x = 180 ÷ 3

x = 60°

(b) What is the maximum exterior angle possible for a regular polygon?

The polygon with the shortest side will be a triangle. Therefore, the value of the maximum external angle will be 120°.

∴ x + x + x = 360°

3x = 360°

x = 360 ÷ 3

x = 120°