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:

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

(b) Simple closed curve

(c) Polygons

(d) Convex polygons

(e) Concave polygon

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.

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

(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.

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

Hence, the sum of measures of the triangles of a convex quadrilateral is 360^o.
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,  1 +  2 +  3 = 180^o ……….(i)
In ΔBDC,  4 +  5 +  6 = 180^o ……….(i)

Adding eq. (i) and (ii),

Hence proved.

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	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 180^o = (7 - 2)x180^o = 5x180^o = 900^o
(b) When n = 8, then
Angle sum of a polygon = (n - 2)x 180^o = (8 - 2)x180^o = 6x180^o =1080^o
(c) When n = 10, then
Angle sum of a polygon = (n - 2)x 180^o = (10 - 2)x180^o = 8x180^o =1440^o
(d) When n = n, then
Angle sum of a polygon = (n - 2)x 180^o

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:

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

50^o+130^o+120^o+ x = 360^o
 300^o+ x = 360^o
 x = 360^o - 300^o
 x = 60^o

(b) Using angle sum property of a quadrilateral,

90^o+60^o+70^o+ x = 360^o
 220^o+ x = 360^o
 x = 360^o-220^o
 x =140^o

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

 Angle sum of a polygon = (n - 2)x180^o
= (5-2)x 180^o = 3x180^o = 540^o

(d) Angle sum of a polygon = (n - 2)x180^o

Hence each interior angle is 108^o

Question 7:
 (a) Find x + y + z

(b) Find x+y+z+w

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

[Exterior angle property]

 x+y+x=90^o+120^o+150^o=360^o

(b) Using angle sum property of a quadrilateral,

Since sum of linear pair angles is 180.
 w+100 +180^o ……….(i)
x+120^o +180^o ……….(ii)
y +80^o +180^o ……….(iii)
z +60^o +180^o ……….(iv)
Adding eq. (i), (ii), (iii) and (iv),
 x + y + z + w+ 100^o +120^o + 80^o + 60^o +180^o +180^o +180^o +180^o
 x + y + z + w+ 360^o = 720^o
 x + y + z + w = 720^o - 360^o
 x + y + z + w = 360^o