# Understanding Quadrilaterals| NCERT Solutions| Class 8

NCERT Solutions of  Chapter: Understanding Quadrilaterals|. NCERT Solutions along with worksheets and notes for Class 8.

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

Ans-

 SN Basis of classification Number of figure (a) Simple curve 1, 2, 5, 6, 7 (b) Simple closed curve 1,2, 5, 6, 7 (c) Polygon 1, 2 (d) Convex polygon 2 (e) Concave polygon 1

(2) How many diagonals does each of the following have?

(a) A convex quadrilateral  (b) A regular hexagon (c) A triangle

Ans-

(a)

AC and BD are two diagonals.

(b) A regular hexagon

ABCDEF is a regular hexagon.

AC, AE, AD, BD, BF, BE, CE, CF and DE are diagonals.

(c) A triangle

ABC is a triangle.

Triangle does not have any diagonals because there are no two non – consecutive vertices. All vertices of triangle are consecutive. So, any two vertices cannot be connected to draw diagonal.

AB, AC and BC are sides of triangle. A, B and C are consecutive vertices of triangle.

(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!)

Ans-

(i)

(5) What is a regular polygon?

State the regular polygon?

State the name of a regular polygon of

(i) 3 sides (ii) 4 sides   (iii) 6 sides

Ans-

The polygons that have equal lengths and equal angles are called regular polygons.

Eg:- An equilateral triangle[It has equal sides and equal angles]

Name of the regular polygon according to their sides

 S.N. Sides of regular polygon Name of the regular polygon (i) 3 Triangle (ii) 4 sides Quadrilateral (iii) 6 sides Hexagon

(6) Find the angle measure x in the following figures:

Ans-

(a)

(b)

(c)

(d)

(7)

(a)Find x + y + z

(b)Find x + y + z + w

Ans-

(a)

(b)

Exercise 3.2

(1) Find x in the following figures.

Ans-

(a)

(b)

(5) (a) Is it possible to have a regular polygon with measure of each exterior angle as 220?

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

Ans-

(a)

(b)

Ans-

(i)

(ii)

(iii)

(iv)

(v)

(i)

(a) It has four sides.

Satisfied.

(b)Its opposite sides are parallel and equal in length.

Donot know

(c)The opposite angles of a parallelogram are of equal measure.

Donot know

(d) The adjacent angles in a parallelogram are supplementary.

Satisfied. A + C and D + B are adjacent and supplementary.

(e) The diagonals of a parallelogram bisect each other at the point of intersection.

Satisfied.

But all conditions for a parallelogram are not satisfied. Hence, it need not be a parallelogram, but it can be.

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

AB = DC = 8cm

BC = 4.4 cm

In a parallelogram opposite sides are equal in length.

But,

AD and BC are opposite and not equal sides.

Hence, it can not be a parallelogram.

(iii) ∠A = 700 and ∠C = 650

Opposite angles ∠A = 700 and ∠C = 650 are not equal.

Hence, it can not be a parallelogram.

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

Ans-

A quadrilateral that is not parallelogram but has exactly two opposite angles of equal measure is kite.

In figure below ABCD is a kite.

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

Ans-

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

Ans-

(8) The following figures GUNS and RUNS are parallelograms. Find x and y.(Lengths are in cm)

Ans-

Ans-

(i)

(ii)

y = 20 -7

y = 13

RO = ON

16 = x + y

Or

x + y = 16

Putting value of y

x + 13 = 16

Transposing 13 to RHS

x = 1 6– 13

x = 3

Hence, x = 3 and y = 13.

(9) In the given figure both RISK and CLUE are parallelograms. Find the value of x.

Ans-

(10) Explain how this figure is a trapezium. Which of its two sides are parallel?

Ans-

Ans-

Ans-

Exercise 3.4

(1) State whether True and False.

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All squares are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

Ans-

(a) All rectangles are squares.   False

(b) All rhombuses are parallelograms.    True

(c) All squares are rhombuses and also rectangles.   True

(d) All squares are not parallelograms.    False

(e) All kites are rhombuses.     False

(f) All rhombuses are kites.   True

(g) All parallelograms are trapeziums.   True

(h) All squares are trapeziums.   True

(2) Identify all the quadrilaterals that have.

(a) four sides of equal length

(b) four right angles

Ans-

(a) four sides of equal length

(i) Square

(ii) Rhombus

(b) four right angles

(i) Square

(ii)Rectangle

(3) Explain how a square is.

(i) a quadrilateral  (ii) a parallelogram   (iii) a rhombus    (iv) a rectangle

Ans-

(i)

(ii)

(iii)

(iv)

(4) Name the quadrilaterals whose diagonals.

(i) bisect each other   (ii) are perpendicular bisectors of each other   (iii) are equal

Ans-

(i) The quadrilaterals whose diagonals bisect each other

(a) Parallelogram

(b) Rhombus

(c) Square

(d)Rectangle

(ii) The quadrilaterals whose diagonals are perpendicular bisectors of each

(a) Rhombus

(b) Square

(iii) The quadrilaterals whose diagonals are equal

(a)Rectangle

(b) Square

(5) Explain why a rectangle is a convex quadrilateral.

Ans-

Let ABCD is a rectangle.

AB, BC, CD and DA are sides of it.

AC and BD are diagonals of it.

Both diagonal lies inside the rectangle.

Hence, a rectangle is a convex quadrilateral.

(6) ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

Helping Topics

Worksheet Class 8