3. Understanding Quadrilaterals – Class 8 Mathematics NCERT Chapter 3 Complete Guide (CBSE)
NCERT Class 8 Mathematics Chapter 3. Understanding Quadrilaterals – All Exercise Solutions | CBSE Board
Chapter 3: Understanding Quadrilaterals
Exercise 3.1
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: (a) Figure (1) , Figure (2) , Figure (3) , Figure (5) , Figure (6) and Figure (7) are simple curve .
(b) Figure (1) , Figure (2) , Figure (3) , Figure (5) , Figure (6) and Figure (7) are simple closed curve .
(c) Figure (1) and Figure (2)are polygon .
(d) Figure (2) is convex polygon .
(e) Figure (1) is concave polygon
2. How many diagonals does each of the following have?
(a) A convex quadrilateral (b) A regular hexagon (c) A triangle
Answer: (a) A convex quadrilateral have two diagonals .
(b) A regular hexagon has nine diagonals .
(c) A triangle have no diagonal .
[The number of diagonal 
, where 
is the number of sides of polygon.]
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: The sum of the measures of the angles of a convex quadrilateral is 360° . Yes .
4. Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)
What can you say about the angle sum of a convex polygon with number of sides?
(a) 7 (b) 8 (c) 10 (d) n
Solution: We know that , the angle sum of a convex polygon 
(a) 900° (b) 1080° (c) 1440° (d) 
5. What is a regular polygon? State the name of a regular polygon of : (i) 3 sides (ii) 4 sides (iii) 6 sides
Answer: A regular polygon is equiangular and equilateral . For example : a square has sides of equal length and angles of equal measures .
(i) Equilateral triangle (for 3 sides) (ii) Square (for 4 sides) (iii) Regular hexagon (for 6 sides)
6. Find the angle measure x in the following figures.
Solution: (a) Given figure,
We know that , the sum of the measures of the four angles of a quadrilateral is 360° .
Therefore, the value of 
is 60° .
Solution: (b) Given figure,
We know that , the sum of the measures of the four angles of a quadrilateral is 360° .
Therefore, the value of is 140° .
Solution: (c) Given figure,
We know tha, 
So, 
Again , 
Therefore, the value of is 140° .
Solution: (d) Given figure,
We have , 
Therefore, the value of is 108° .
7.
Solution: (a) Given figure,
We know that , Interior angle+Exterior angle=180°
So, 
Again , 
The sum of the three interior angle of the polygon is 180° .
And 
Solution: (b) Given, figure,
We know that ,
We know that , the sum of the measures of the four angles of a quadrilateral is 360° .
And 
Exercise 3.2
1. Find x in the following figures.
Solution: (a) Given figure ,
We know that , the sum of the measure of the external angles of any polygon is 360° .
Therefore, the value of is 110° .
Solution: (b) Given figure,
We know that , the sum of the measure of the external angles of any polygon is 360° .
Therefore, the value of is 50° .
2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides
Solution: We know that, the measure of each external angle 
, where is the number of sides .
(i) the measure of each exterior angle of a regular 
(ii) the measure of each exterior angle of a regular 
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
Solution: We know that , the number of sides of a regular polygon 
4. How many sides does a regular polygon have if each of its interior angles is 165°?
Solution: We know that, interior angle + exterior angle = 180°
Exterior angle = 180° – 165° = 15°
Therefore, the number of sides of a regular polygon 
5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°? (b) Can it be an interior angle of a regular polygon? Why?
Solution: (a) We know that , the number of sides of a regular polygon 
No . because 22° is not divisor of 360° .
(b) No , Because Exterior angle = 180° – 22° = 158° is not a divisor of 360° .
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 minimum interior angle possible for a regular polygon is an equilateral triangle . Because , the equilateral triangle being a regular polygon of 3 sides has the least measure of an interior angle = 60° .
(b) The maximum exterior angle possible for a regular polygon is 120° .
Exercise 3.3
1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = ...... (ii) ∠ DCB = ......
(iii) OC = ...... (iv) m ∠DAB + m ∠CDA = ......
Solution: (i) Since , the opposite sides of a parallelogram are of equal length .
So , AD = BC
(ii) Since, the opposite angles of a parallelogram are of equal measure .
So, 
(iii) Since, the diagonal of a parallelogram bisect each other .
So, OC = OA
(iv) Since , the adjacent angles in a parallelogram are supplementary .
So, 
2. Consider the following parallelograms. Find the values of the unknowns x, y, z.
Solution: (i) Since , the adjacent angles in a parallelogram are supplementary .
Again , the opposite angles of a parallelogram are of equal measure .
and 
Thus , the value of 
, 
and 
(ii) Since , the adjacent angles in a parallelogram are supplementary .
Again , the opposite angles of a parallelogram are of equal measure .
and 
[ Corresponding angles are equal]
Thus, the value of 
, 
and 
.
(iii) We have, 
[Vertically opposite angle]
Since , the sum of all the angle of triangle is 180° .
[Alternative interior angle]
Thus, the value of 
, 
and 
.
(iv) We have , [Corresponding angle are equal]
The opposite angles of a parallelogram are of equal measure .
The adjacent angles in a parallelogram are supplementary .
Therefore, the value of 
, and .
(v) We have, 
[Opposite angles are equal]
The adjacent angles in a parallelogram are supplementary .
[Alternative interior angles are equal]
Therefore, the value of 
, and 
.
3. Can a quadrilateral ABCD be a parallelogram if
(i) 
? (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) 
and 
?
Solution: (i)
ABCD quadrilateral may be a parallelogram , if the condition of the parallelogram are fulfilled .
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm
Since , the opposite sides of a parallelogram are equal .
So, 
, but 
.
(iii) and
Since , the opposite angles of a parallelogram are equal .
But 
.
4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Solution: A quadrilateral that is not a parallelogram but has exactly two opposite angles equal can be drawn as follows:
Here, ∠A = ∠C, but the other pair of opposite angles is not equal. Also, the opposite sides are not parallel.
Therefore, ABCD is a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
5. The measures 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 
and 
be the angles of the parallelogram.
Since, the adjacent angles in a parallelogram are supplementary .
Thererfore, the angle are : 
and 
6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.
Solution: Let be the angles of a parallelogram .
Since, the adjacent angles in a parallelogram are supplementary .
Therefore, the measure of the angles of the parallelogram are : 90° and 90° .
7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
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