NCERT Solution for class 8 maths chapter-3 Understanding Quadrilaterals

NCERT solutions for class 8 maths chapter-3 Understanding Quadrilaterals is prepared by academic team of Physics Wallah. we have prepared solutions for all exercise of chapter-3. Given below is step by step solutions of all questions given in NCERT textbook for chapter-3. Also read chapter-3 theory make sure you have gone through the theory part of chapter-3 from NCERT textbook and you have learned the formula of the given chapter. Physics Wallah prepared a detail notes and additional questions for class 8 maths with short notes of all maths formula of class 8 maths. Do read these contents before moving to solve the exercise of NCERT chapter-3

NCERT Solutions for Class 8 Maths Exercise 3.1


Question 1.
Given here are some figures:
Understanding Quadrilaterals/image001.png
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
Understanding Quadrilaterals/image002.jpg
(b) Simple closed curve

Question 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 has two diagonals.
Here, AC and BD are two diagonals.
Understanding Quadrilaterals/image007.jpg
(b) A regular hexagon has 9 diagonals.
Here, diagonals are AD, AE, BD, BE, FC, FB, AC, EC and FD.
Understanding Quadrilaterals/image008.jpg
(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)
Solution :
Let ABCD is a convex quadrilateral, then we draw a diagonal AC which divides the quadrilateral in two triangles.
∠A + ∠B + ∠C + ∠D
= ∠1 + ∠6 + ∠5 + ∠4 + ∠3 + ∠2
= (∠1 + ∠2 + ∠3) + (∠4 + ∠5 + ∠6)
Understanding Quadrilaterals/image010.jpg

[By Angle sum property of triangle]
= 360º
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.
Understanding Quadrilaterals/image014.jpg
Using angle sum property of triangle,
In △ABD, ∠1 + ∠2 + ∠3 = 180º;……….(i)
In △BDC, ∠4 + ∠5 + ∠6 = 180º;……….(i)
Adding eq. (i) and (ii),
∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360º
⇒;∠1 + ∠2 + (∠3 + ∠4) + ∠5 + ∠6
= 360º
⇒;∠A + ∠B + ∠C + ∠D = 360º
Hence proved.
Question 4.
Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
1.jpg
Solution :
(a) When n = 7, then
Angle sum of a polygon = Understanding Quadrilaterals/image027.png
Understanding Quadrilaterals/image028.png
(b) When n= 8, then
Angle sum of a polygon = Understanding Quadrilaterals/image027.png
Understanding Quadrilaterals/image029.png
(c) When n = 10, then
Angle sum of a polygon = Understanding Quadrilaterals/image027.png
Understanding Quadrilaterals/imageimage028.png
(d) When n = n then
Angle sum of a polygon = Understanding Quadrilaterals/image027.png
Question 5.
What is a regular polygon? State the name of a regular polygon of:
(a) 3 sides
(b) 4 sides
(c) 6 sides
Solution :
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.
https://farm8.staticflickr.com/7255/7814800658_405d4947d4_o.png


 x = 140º

 (d)


NCERT Solutions for Class 8 Maths Exercise 3.2


Question 1.
Find x;in the following figures:
Understanding Quadrilaterals/image002.png
Solution :
(a) Here, Understanding Quadrilaterals/image003.png
[Linear pair]
Understanding Quadrilaterals/image004.jpg

Understanding Quadrilaterals/image016.jpg
By linear pairs of angles,
Understanding Quadrilaterals/image017.png

Adding eq. (i), (ii), (iii), (iv) and (v),
Understanding Quadrilaterals/image022.png
 


Question 2.
Find the measure of each exterior angle of a regular polygon of:
(a) 9 sides
(b) 15 sides
Solution :
(i) Sum of angles of a regular polygon = Understanding Quadrilaterals/image013.png
= Understanding Quadrilaterals/image026.png
Each interior angle = Understanding Quadrilaterals/image027.png
Each exterior angle = Understanding Quadrilaterals/image028.png
(ii) Sum of exterior angles of a regular polygon = 360º
Each interior angle = Understanding Quadrilaterals/image030.png


Question 3.
How many sides does a regular polygon have, if the measure of an exterior angle is 24º?


Solution :
Let number of sides be  n.
Sum of exterior angles of a regular polygon = 360º
Number of sides = Understanding Quadrilaterals/image033.png
Hence, the regular polygon has 15 sides.


Question 4.
How many sides does a regular polygon have if each of its interior angles is  165º?
Solution :
Let number of sides be n.
Exterior angle = Understanding Quadrilaterals/image035.png
Sum of exterior angles of a regular polygon = 360º
Number of sides = Understanding Quadrilaterals/image036.png
Hence, the regular polygon has 24 sides.
Question 5.
(a) Is it possible to have a regular polygon with of each exterior angle as 22º?
(b) Can it be an interior angle of a regular polygon? Why?
Solution :
(a) No. (Since 22 is not a divisor of 360º)
(b) No, (Because each exterior angle is 180º -  22º = 158º; which is not a divisor of 360º)
 

Question 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) The equilateral triangle being a regular polygon of 3 sides has the least measure of an
interior angle of 60º,
∵ Sum of all the angles of a triangle
= 180º
∴;   x + x +x = 180º
    ⇒     3 x = 180º
        ⇒       x  = 60º
(b) By (a), we can observe that the greatest exterior angle is   180º -  60º
= 120º.


NCERT Solutions for Class 8 Maths Exercise 3.3


Question 1.
Given a parallelogram ABCD. Complete each statement along with the definition or property used.
Understanding Quadrilaterals/image001.jpg
(i) AD = _______________
(ii) ∠DCB = ______________
(iii) OC = _____________
(iv) m∠DAB + m∠CDA = ________


Solution :
(i) AD = BC
[Since opposite sides of a parallelogram are equal]
(ii) ∠DCB = ∠DAB
[Since opposite angles of a parallelogram are equal]
(iii) OC = OA
[Since diagonals of a parallelogram bisect each other]
(iv) m∠DAB + m∠CDA = 180º
[Adjacent angles in a parallelogram are supplementary]


Question 2.
Consider the following parallelograms. Find the values of the unknowns x , y ,z
Understanding Quadrilaterals/image006.png

Understanding Quadrilaterals/image009.png

Note: For getting correct answer, read  3º  =  30º in figure (iii)
Solution :
(i) ∠B + ∠C = 180º

[Adjacent angles in a parallelogram are supplementary]
Understanding Quadrilaterals/image012.jpg
Understanding Quadrilaterals/image014.png

[Since opposite angles of a parallelogram are equal]
Also   y = 100º
[Since opposite angles of a parallelogram are equal]
(ii)  x + 50º = 180º x = 50º = 180º
[Adjacent angles in a ||gm are supplementary]
Understanding Quadrilaterals/image020.jpg
 Understanding Quadrilaterals/image013.png
[Corresponding angles]
(iii)  x = 90º
[Vertically opposite angles]
Understanding Quadrilaterals/image024.jpg
Understanding Quadrilaterals/image025.png

[Adjacent angles in a ||gm are supplementary]
Understanding Quadrilaterals/image032.jpg

[Opposite angles are equal in a ||gm]
(v)  y = 112º
[Opposite angles are equal in a Understanding Quadrilaterals/image019.pnggm]
Understanding Quadrilaterals/image036.jpg
[Alternate angles]


Question 3.


Can a quadrilateral ABCD be a parallelogram, if:
(i) ∠D + ∠B = 180º
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) ∠A = 70º and ∠C = 65º?


Solution :
(i) ∠D + ∠B = 180º
It can be, but here, it needs not to be.
Understanding Quadrilaterals/image045.jpg
(ii) No, in this case because one pair of opposite sides are equal and another pair of opposite sides are unequal. So, it is not a parallelogram.
Understanding Quadrilaterals/image046.jpg
(iii) No.Understanding Quadrilaterals/image002.png
Since opposite angles are equal in parallelogram and here opposite angles are not equal in quadrilateral ABCD. Therefore it is not a parallelogram.
Understanding Quadrilaterals/image048.jpg


Question 4.


Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measures.


Solution :
ABCD is a quadrilateral in which angles Understanding Quadrilaterals/image002.pngA = Understanding Quadrilaterals/image002.pngC = 110º
Therefore, it could be a kite.
Understanding Quadrilaterals/image050.jpg



Question 5.


The measure of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.


Solution :
Let two adjacent angles be 3X and 2X
Understanding Quadrilaterals/image053.jpg
Since the adjacent angles in a parallelogram are supplementary.
Understanding Quadrilaterals/image055.png
 


Question 6.

Two adjacent angles of a parallelogram have equal measure. Find the measure of the angles of the parallelogram.


Solution : Let each adjacent angle be x,
Since the adjacent angles in a parallelogram are supplementary.
Understanding Quadrilaterals/image061.png

Hence, each adjacent angle is 90º


Question 7.
The adjacent figure HOPW is a parallelogram. Find the angle measures  x , y and z State the properties you use to find them.
Understanding Quadrilaterals/image067.jpg
∠HOP +  70º = 180º


 


Question 8.
The following figures GUNS and RUNS are parallelograms. Find x and y (Lengths are in cm)
Understanding Quadrilaterals/image079.jpg

Solution :


(i) In parallelogram GUNS,
GS = UN
[Opposite sides of parallelogram are equal]
Understanding Quadrilaterals/image081.png

Also GU = SN
[Opposite sides of parallelogram are equal]
Understanding Quadrilaterals/image083.png

Hence, x= 6 cm and x= 9 cm.
(ii) In parallelogram RUNS,
Understanding Quadrilaterals/image088.png
 ⇒ x = 16 -13

 ⇒ x = 16 -13

⇒ x = 3cm


Hence,  x = 3 cm and y = 13cm.


Question 9.
In the figure, both RISK and CLUE are parallelograms. Find the value of Understanding Quadrilaterals/image060.png
Understanding Quadrilaterals/image095.png
Solution :


In parallelogram RISK,


 Understanding Quadrilaterals/image097.png

[Vertically opposite angles]


Question 10.


Explain how this figure is a trapezium. Which is its two sides are parallel?
Understanding Quadrilaterals/image106.jpg
Solution :
Here, ∠M + ∠L =  100º + 80º = 180º
[Sum of interior opposite angles is 180º]
NM and KL are parallel.
Hence, KLMN is a trapezium.

Understanding Quadrilaterals/image110.jpg
Solution :

Understanding Quadrilaterals/image112.png
 


Question 12.
Find the measure of ∠P and ∠S if Understanding Quadrilaterals/image116.pngin given figure.
(If you find m∠R is there more than one method to find m∠P)
Understanding Quadrilaterals/image117.jpg
Solution :



  Understanding Quadrilaterals/image123.png
 


NCERT Solutions for Class 8 Maths Exercise 3.4


2.jpg


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