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

NCERT Solutions Class 8th maths Chapter 3 Understanding of Quadrilaterals. 8th class mathematics. Here We learn what is in ncert class 8 maths solutions Chapter 3 and solved questions with easiest method. In this chapter we solve the question of ncert 8th class mathematics Chapter 3 exercise 8.1 class 8 maths, class 8 maths Chapter 3 exercise 3.1, class 8 maths Chapter 3 exercise 3.2, class 8 maths Chapter 3 ex 3.3 class 8 and class 8 maths Chapter 3 exercise 3. class 8th maths
NCERT Solutions for Class 8 Math Chapter 3 are part of ncert solutions for class 8 maths PDF. Ncert solutions for class 8 maths. Ncert solutions for class 8 Chapter 3 maths Chapter 3 Understanding of Quadrilaterals with formula and solution.

Here we solve ncert class 8th maths solutions Chapter 3 Understanding of Quadrilateral with formula and solution concepts all questions with easy method with expert solutions. It helps students in their study, homework and preparing for exam. Soon we provide NCERT class 8 Maths Chapter 3 Chaturbhujon ko Samajhna question and answers. Soon we provided ncert solutions for ncert class 8 maths Chapter 3 Understanding of Quadrilaterals in free PDF here. ncert class 8 maths solutions pdf will be provide soon. ch8 maths class 8 NCERT Solution and book PDF.

## NCERT SOLUTIONS FOR CLASS 8 MATHS CHAPTER 3 Understanding of Quadrilaterals 8th class mathematics Ex 3.1

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.)

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°

Therefore, 150° + 90° + a = 360°
240° + a = 360°
a = 360° – 240°
a = 120°
So, x + y + z = 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°

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