Class 8 Exam  >  Class 8 Notes  >  NCERT Solution (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths, Class 8

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Understanding Quadrilaterals

Exercise 3.1

Question 1:
 Given here are some figures:

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

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

Answer 1:

(a) Simple curve

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(b) Simple closed curve

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(c) Polygons

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(d) Convex polygons

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(e) Concave polygon

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Question 2:
 How many diagonals does each of the following have?
 (a) A convex quadrilateral
 (b) A regular hexagon
 (c) A triangle

Answer 2:
(a) A convex quadrilateral has two diagonals.
Here, AC and BD are two diagonals.

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(c) A triangle has no diagonal.

Question 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)
 Answer 3:

Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

= 180+180o                              [By Angle sum property of triangle]
= 360o

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
Hence, the sum of measures of the triangles of a convex quadrilateral is 360o.
Yes, if quadrilateral is not convex then, this property will also be applied.

Let ABCD is a non-convex quadrilateral and join BD, which also divides the quadrilateral in two triangles.
Using angle sum property of triangle,
In ΔABD, NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 1 + NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 2 + NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 3 = 180o ……….(i)
In ΔBDC, NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 4 + NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 5 + NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 6 = 180o ……….(i)

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Adding eq. (i) and (ii),

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Hence proved.

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

FigureNCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, MathsNCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, MathsNCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, MathsNCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Side

3

4

5

6

Angle sum

1 x 1800 = (3- 2) x 1800

2x1800 = (4-2)xl800

3x1800 = (5-2)xl800

4x1800 = (6-2)xl800

What can you say about the angle sum of a convex polygon  with number of sides?

Answer 4:
(a) When n = 7, then
Angle sum of a polygon = (n - 2)x 180o = (7 - 2)x180o = 5x180o = 900o
(b) When n = 8, then
Angle sum of a polygon = (n - 2)x 180o = (8 - 2)x180o = 6x180o =1080o
(c) When n = 10, then
Angle sum of a polygon = (n - 2)x 180o = (10 - 2)x180o = 8x180o =1440o
(d) When n = n, then
Angle sum of a polygon = (n - 2)x 180o

Question 5:
 What is a regular polygon? State the name of a regular polygon of:
 (a) 3 sides
 (b) 4 sides
 (c) 6 sides

Answer 5:
A regular polygon: A polygon having all sides of equal length and the interior angles of equal size is known as regular polygon.
(i) 3 sides - Polygon having three sides is called a triangle.
(ii) 4 sides - Polygon having four sides is called a quadrilateral.
(iii) 6 sides - Polygon having six sides is called a hexagon.

Question 6:
 Find the angle measures x in the following figures:

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

 

Answer 6:
(a) Using angle sum property of a quadrilateral,

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

50o+130o+120o+ x = 360o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 300o+ x = 360o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x = 360- 300o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x = 60o

(b) Using angle sum property of a quadrilateral,

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
90o+60o+70o+ x = 360o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths 220o+ x = 360o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x = 360o-220o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x =140o

(c) First base interior angle = 180o- 70o =110o
Second base interior angle = 180o- 60o = 120o
There are 5 sides, n = 5

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths Angle sum of a polygon = (n - 2)x180o
= (5-2)x 180= 3x180= 540o

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(d) Angle sum of a polygon = (n - 2)x180o

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Hence each interior angle is 108o

Question 7:
 (a) Find x + y + z

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

(b) Find x+y+z+w

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Answer 7:
(a) Since sum of linear pair angles is 180o.

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

[Exterior angle property]

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x+y+x=90o+120o+150o=360o

(b) Using angle sum property of a quadrilateral,

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

Since sum of linear pair angles is 180.
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths w+100 +180o ……….(i)
x+120o +180o ……….(ii)
y +80o +180o ……….(iii)
z +60o +180o ……….(iv)
Adding eq. (i), (ii), (iii) and (iv),
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x + y + z + w+ 100o +120o + 80o + 60o +180o +180o +180o +180o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x + y + z + w+ 360o = 720o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x + y + z + w = 720o - 360o
NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths x + y + z + w = 360o

 

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FAQs on NCERT Solutions for Class 8 Maths - (Ex - 3.1) - Chapter 3: Understanding Quadrilaterals, Maths

1. What are quadrilaterals?
Ans. Quadrilaterals are polygons that have four sides, four angles, and four vertices.
2. What are the types of quadrilaterals?
Ans. The types of quadrilaterals are square, rectangle, parallelogram, rhombus, trapezium, kite, and irregular quadrilateral.
3. What is the difference between a parallelogram and a rhombus?
Ans. A parallelogram has opposite sides parallel and equal in length, while a rhombus has all four sides equal in length.
4. How do we determine if a quadrilateral is a kite?
Ans. A kite is a quadrilateral with two pairs of adjacent sides that are of equal length. To determine if a quadrilateral is a kite, we need to check if it has two pairs of adjacent sides of equal length.
5. What is the sum of interior angles of a quadrilateral?
Ans. The sum of interior angles of a quadrilateral is 360 degrees.
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