Ncert solution for class 8 chapter 3 understanding quadrilaterals exercise 3.1

Exercise 3.1

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Introduction - Ex 3.1 Understanding Quadrilaterals - NCERT Class 8th Maths Solutions

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Q 1 - Ex 3.1 - Understanding Quadrilaterals - NCERT Maths Class 8th - Chapter 3

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Question 1
Given here are some figures.

(1)	(2)	(3)
(4)	(5)	(6)
(7)	(8)

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
Sol :
(a) 1, 2, 5, 6, 7
(b) 1, 2, 5, 6, 7
(c) 1, 2
(d) 2
(e) 1

Q 2 - Ex 3.1 - Understanding Quadrilaterals - NCERT Maths Class 8th - Chapter 3

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Question 2
How many diagonals does each of the following have?
(a) A convex quadrilateral
Sol :
There are 2 diagonals in a convex quadrilateral.

(b) A regular hexagon
Sol :
There are 9 diagonals in a regular hexagon.


(c) A triangle
Sol :
A triangle does not have any diagonal in it.

Q 3 - Ex 3.1 - Understanding Quadrilaterals - NCERT Maths Class 8th - Chapter 3

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Question 3
What is the sum of the measures of the angels of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
Sol :
The sum of the measures of the angles of a convex quadrilateral is 360° as a convex quadrilateral is made of two triangles.

Here, ABCD is a convex quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, the sum of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles i.e., 180º + 180º = 360º
Yes, this property also holds true for a quadrilateral which is not convex. This is because any quadrilateral can be divided into two triangles.

Here again, ABCD is a concave quadrilateral, made of two triangles ΔABD and ΔBCD. Therefore, sum of all the interior angles of this quadrilateral will also be 180º + 180º = 360º

Q 4 - Ex 3.1 - Understanding Quadrilaterals - NCERT Maths Class 8th - Chapter 3

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Question 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	180°	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
Sol :
From the table, it can be observed that the angle sum of a convex polygon of n sides is (n −2) × 180º. Hence, the angle sum of the convex polygons having number of sides as above will be as follows.
(a) (7 − 2) × 180º = 900°
(b) (8 − 2) × 180º = 1080°
(c) (10 − 2) × 180º = 1440°
(d) (n − 2) × 180°

Q 5 - Ex 3.1 - Understanding Quadrilaterals - NCERT Maths Class 8th - Chapter 3

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Question 5
What is a regular polygon?
State the name of a regular polygon of
(i) 3 sides
(ii) 4 sides
(iii) 6 sides
Sol :
A polygon with equal sides and equal angles is called a regular polygon.
(i) Equilateral Triangle

(ii) Square

(iii) Regular Hexagon

Q 6 - Ex 3.1 -Class 8th - Chapter 3 Understanding Quadrilaterals - NCERT Maths 

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

(a)	(b)
(c)	(d)

Sol :
(a)
Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,
50° + 130° + 120° + x = 360°
300° + x = 360°
x = 60°
(b)

From the figure, it can be concluded that,
90º + a = 180º (Linear pair)
a = 180º − 90º = 90º
Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,
60° + 70° + x + 90° = 360°
220° + x = 360°
x = 140°
(c)

From the figure, it can be concluded that,
70 + a = 180° (Linear pair)
a = 110°
60° + b = 180° (Linear pair)
b = 120°
Sum of the measures of all interior angles of a pentagon is 540º.
Therefore, in the given pentagon,
120° + 110° + 30° + x + x = 540°
260° + 2x = 540°
2x = 280°
x = 140°
(d)
Sum of the measures of all interior angles of a pentagon is 540º.
5x = 540°
x = 108°

Q 7 - Ex 3.1 -Class 8th - Chapter 3 Understanding Quadrilaterals - NCERT Maths 

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Question 7
(a) find x + y + z

Sol :
(a) x + 90° = 180° (Linear pair)
x = 90°
z + 30° = 180° (Linear pair)
z = 150°
y = 90° + 30° (Exterior angle theorem)
y = 120°
x + y + z = 90° + 120° + 150° = 360°

(b) find x + y + z + w






Sol :
Sum of the measures of all interior angles of a quadrilateral is 360º. Therefore, in the given quadrilateral,
a + 60° + 80° + 120° = 360°
a + 260° = 360°
a = 100°
x + 120° = 180° (Linear pair)
x = 60°
y + 80° = 180° (Linear pair)
y = 100°
z + 60° = 180° (Linear pair)
z = 120°
w + 100° = 180° (Linear pair)
w = 80°
Sum of the measures of all interior angles = x + y + z + w
= 60° + 100° + 120° + 80°
= 360°