NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

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

Ans : The sum of the measures of the angles of a convex quadrilateral is 3600 as a convex quadrilateral is made of two triangles. 

Here, ABCD is a convex quadrilateral, made Of two triangles \(\triangle A B D\) and \(\triangle B C D\).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., \(180^{\circ}+180^{\circ}=360^{\circ}\) 
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 \(\triangle A B D\) and \(\triangle B C D\) .
Therefore, sum of all the interior angles of this quadrilateral will also be \(180^{\circ}+180^{\circ}=360^{\circ}\). 

Ans : From the table, it can be observed that the angle sum of a convex polygon of n sides is (n —2) x \(180^{\circ}\). Hence, the angle sum of the convex polygons having number of sides as above will be as follows. 
(a) (7 - 2) x  \(180^{\circ}\) = \(900^{\circ}\)
(b) (8 - 2) x  \(180^{\circ}\) =  \(1800^{\circ}\) 
(c) (10 - 2) x  \(180^{\circ}\) =  \(180^{\circ}\) 
(d) (n - 2) x  \(180^{\circ}\) 

Ans : A polygon with equal sides and equal angles is called a regular polygon. 
 

Ans : (a) Sum of the measures of all interior angles of a quadrilateral is \(360^{\circ}\)  Therefore, in the given quadrilateral, 
\(50^{\circ}\) + \(130^{\circ}\) + \(120^{\circ}\) + x = \(360^{\circ}\) 
\(300^{\circ}\) + x = \(360^{\circ}\) 
x = \(60^{\circ}\)


(b) 

From the figure, it can be concluded that, 
\(90^{\circ}\) + a = \(180^{\circ}\) (Linear pair) 
a = \(180^{\circ}\) - \(90^{\circ}\) = \(90^{\circ}\) 
Sum Of the measures Of all interior angles of a quadrilateral is \(360^{\circ}\). Therefore, in the given quadrilateral, 
\(60^{\circ}\) + \(70^{\circ}\) + x + \(90^{\circ}\) = \(360^{\circ}\) 
\(220^{\circ}\) + x = \(360^{\circ}\) 
x = \(140^{\circ}\)


(c) 

From the figure, it can be concluded that, 
70 + a = \(180^{\circ}\) (Linear pair) 
a = \(110^{\circ}\) 
\(60^{\circ}\) + b = \(180^{\circ}\) (Linear pair) 
b = \(120^{\circ}\)
Sum Of the measures Of all interior angles Of a pentagon is  \(540^{\circ}\), 
Therefore, in the given pentagon, 
 \(120^{\circ}\) +  \(110^{\circ}\) +  \(30^{\circ}\) + x + x =  \(540^{\circ}\)
 \(260^{\circ}\) + 2x =  \(540^{\circ}\)
2x =  \(280^{\circ}\)
X =  \(140^{\circ}\)


(d) Sum of the measures of all interior angles of a pentagon is  \(540^{\circ}\) 
5x =  \(540^{\circ}\) 
x =  \(108^{\circ}\)