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

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

The classification of the given figures is as under :

(a) (1), (2), (5), (6), (7) and (8) are Simple curves.

(b) (1), (2), (5), (6) and (7) are Simple closed curves.

(c)(1) and (2) are Polygon.

(d)(2) is Convex polygon.

(e) (1) and (4) are Concave polygons.

2.How many diagonals does each of the following have?

(a) A convex quadrilateral

(b) A regular hexagon

(c) A triangle.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(a)

A convex quadrilateral has two diagonals.

(b)

A regular hexagon has nine diagonals.

(c)

A triangle does not have any diagonal.

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 j try!)

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

The sum of measures of the angles of a convex quadrilateral is

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

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

From the given table, dearly we observe that the sum of angles (interior angles) of a polygon with n sides =

(a) n = 7

Sum of angle =

?The sum of angle for 7 sided polygon=

(b) n = 8

Sum of angles =

? The sum of the angles of a polygon of 8 sides=

(c) n = 10

Angle sum =

? The sum of the angles of a polygon of 10 sides=

(d) We can observe from the given table that the number of triangles is two less them the number of sides in the polygon.

? If the polygon has n sides, the number of triangles formed will be (n – 2).

Also we know that the sum of angles of a triangle =

? The sum of angles of a polygon of n sides =

5.What is a regular polygon? State the name of a regular polygon of

(i) 3 slides

(ii) 4 slides

(iii) 6 slides

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

A polygon is said to be a regular polygon, if all its sides and interior angles and exterior angles are equal.

The regular polygon of:

(i)3 sides= equilateral triangle.

(ii)4 sides= square.

(iii)6 sides = regular hexagon

.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(a) As we know the sum of interior angles of a quadrilateral is

?

(b) As we know the sum of interior angles of a quadrilateral is

?

(c) As we see the given figure with 5 sides, so the sum of interior angles of this polygon is =

So in the figure we have ,

and also,

Therefore, the sum of internal angles:

(d) As we see the given figure with 5 sides, so the sum of interior angles of this polygon is =

Therefore, the sum of internal angles:

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(a)We know that the sum of interior angles of a triangle is

So, we have

So we have,

Similarly,

And also,

So,

(b)We know that the sum of interior angles of a quadrilateral is \(360^{\circ\}\).

Therefore

So we have,

Similarly,

And also,

And also,

So,

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

As we know that the sum of the exterior angles formed by producing the sides of a convex polygon in the same order is equal to 360°. Therefore here we have,

(a)

(b)

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(i) We know that the measure of each exterior angle of a regular polygon =

So, the measure of each exterior angle of 9 sided regular polygon =

(ii) Similarly,the measure of each exterior angle of a regular polygon =

So, the measure of each angle of 15 sided regular polygon =

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

We know that sum of all exterior angles of a regular polygon =

So we have,Number of sides=

Hence, the number of sides of the regular polygon is= 15

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

Let the number of sides of a regular polygon be n.

We have, Sum of all interior angles =

and, measure of its each angle=

here we have each angle as

So, we have the number of sides of regular polygon as 24.

5.(a) Is it possible to have a regular polygon with measure of each exterior angle a is

(b) Can it be an interior angle of a regular polygon? Why?

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(a) We have, Number of sides of regular polygon=

As we know number of sides by the formula should be an integer so, it is not possible that a regular polygon have its exterior angle of

(b) If interior angle =

And we have: measure of each interior angle=

But 158 does not divide 360 exactly.So, the polygon is not possible.

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?

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(a) A regular polygon of 3 sides i.e, equilateral triangle has the least measure of an interior angle=

(b) Since the minimum interior angle of a regular polygon is equal to 60°, therefore, the maximum exterior angle possible for a regular polygon =

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 = ……..

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(i) AD = BC as the opposite sides of a parallelogram are equal.

(ii)?DCB = ?DAB as the opposite angles of a parallelogram are equal.

(iii) OC = OA as the diagonals of a parallelogram bisect each other.

(iv) m?DAB + m?CDA =

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(i)Here, ABCD is a parallelogram.

(ii)We have PQRS as a parallelogram.

(iii) Here we have, ABCD as a rhombus because diagonals intersect at

So,now in ?OCB we have,

(iv) ABCD is a parallelogram

?A + ?B = 180^{\circ} (Adjacent angles of a parallelogram are supplementary)

(v) In this case we have ABCD as a parallelogram.

3.Can a quadrilateral ABCD be a parallelogram if

(i)

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii)

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(i) For ?D + ?B =

(ii) Given: AB = DC = 8 cm, AD = 4 cm, BC = 4.4 cm

In general the opposite sides od a parallelogram is equal.

But here,

Thus, ABCD cannot be a parallelogram.

(iii)

Since

Hence, ABCD is not a parallelogram.

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

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

In taking a kite as reference we have exactly two opposite angles of equal measure but not a parallelogram.

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.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

As per the given conditions, let ABCD be parallelogram such that

m?B : m?C = 3 : 2

So,let m?B =

m?B + m?C =

Thus,

and

Hence, the angles of the parallelogram are

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

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

Let ABCD be a parallelogram in which we have,
?A = ?B

We know

Thus,

7.The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

We have,

In ?EPH,

Hence we have,

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals

Answer :

(i)Opposite sides of a parallelogram is equal so we have,

GU=SN

Similarly, we have GS=UN