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NCERT Solutions for class 6 Maths chapter 5 – Understanding Elementary Shapes


Back Exercise

Exercise 5.1

Question 1.
What is the disadvantage in comparing line segments by mere observation?
Solution :
Sometimes the difference in lengths betweenthe two line segments is not obvious. So, we are not always sure about our usual judgment.

Question 2.
Why is it better to use a divider than with a ruler, while measuring the length of a line segment?
Solution :
There may be errors due to the thickness of the ruler and angular viewing by using a ruler. These errors are eradicated by using a divider. So, it is better to use a divider, than a ruler, while measuring the length of a line segment.

Question 3.
Draw any line segment, say AB¯¯¯¯¯¯¯¯. Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A, B, C are any three points on a line such that AC + CB = AB, then we can be sure that C lies between A and B.]
Solution :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 01
Length of AB = 7 cm
Length of BC = 3 cm
Length of AC = 4 cm
AC + CB = 4 cm + 3 cm = 7 cm
But AB = 7 cm
So, AB = AC A- CB.

Question 4.
If A, B, C are three points on a line such that AB = 5 cm, BC – 3 cm and AC – 8 cm, which one of them lies between the other two?
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 2
Solution :
AB + BC = AC, so, the point B lies between the point A and point C.

Question 5.
Verify, whether D is the mid-point of AG.
Solution :
AD = AB + BC + CD = 3 units
DG = OE + EF + FG = 3 units
∴ Yes ! D is the mid-point of AG.
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 3
Question 6.
If B is the mid-point of AC¯¯¯¯¯¯¯¯ and C is the mid-point of BD¯¯¯¯¯¯¯¯, where A, B, C, D lie on a straight line, say why AB = CD ?
Solution :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 4
∴ B is the mid-point of AC¯¯¯¯¯¯¯¯
∴ AB = BC …(1)
∴ C is the mid-point of BD¯¯¯¯¯¯¯¯
∴ BC = CD … (2)
In view of (1) and (2), we get AB = CD.

Question 7.
Draw five triangles and measure their sides. Check in each case, if the sum of the lengths of any two sides is always less than the third side.
Solution :
(i) AB = 3.7 cm
BC = 3 cm
AC = 3.8 cm
tiwari academy class 6 maths Chapter 5 Understanding Elementary Shapes 5
Clearly, AB + BC > AC
BC + AC > AB
AC + AB > BC

(ii) AB = 3 cm
BC = 3 cm
CA = 3 cm
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 6
Clearly, AB + BC > AC
BC + AO AB
AC + AB > B C

(iii) AB = 4 cm
BC = 3 cm
AC = 5 cm
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 7
Clearly, AB + BC > AC
BC+ AC > AB
AC + AB > BC.

(iv) AB = 2 cm
BC = 2 cm
AC = 2.8 cm
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 8
Clearly, AB + BC > AC
BC + AC > AB
AC +AB> BC

(v) AB = 3 cm
BC = 4 cm
CA = 3 cm
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 9
Clearly, AB + BC > AC
BC + AC> AB
AC + AB> BC
In each case, we observe that the sum of the lengths of any two sides is always greater than the third side.

Exercise 5.2

Question 1.
What fraction of a clockwise revolution does the hour hand of a clock turn through, when it goes from
(a) 3 to 9
(b) 4 to 7
(c) 7 to 10
(d) 12 to 9
(e) 1 to 10
(f) 6 to 3?
Solution :
(a) 12 of a revolution
(b) 14 of a revolution
(c) 14 of a revolution
(d) 34 of a revolution
(e) 34 of a revolution
(f) 34 of a revolution.

Question 2.
Where will the hand of a clock stop, if it
(a) starts at 12 and makes 1/2 of a revolution, clockwise?
(b) starts at 2 and makes 1/2 of a revolution, clockwise?
(c) starts at 5 and makes 1/4 of a revolution, clockwise?
(d) starts at 5 and makes 3/4 of a revolution, clockwise?
Solution :
(a) at 6
(b) at 8
(c) at 8
(d) at 2.

Question 3.
Which direction will you face if you start facing
(a) east and make 1/2 of a revolution clockwise?
(b) east and make 1/2 of a revolution clockwise?
(c) west and make 3/4 of a. revolution anti-clockwise?
(d) south and make one full revolution? (Should we specify clockwise or anti-clockwise for this last question? Why not?)
Solution :
(a) West
(b) West
(c) North
(d) South.
To answer (d), it is immaterial whether we turn clockwise or anti-clockwise because one full revolution will bring us back to the original position.

Question 4.
What part of a revolution have you turned through if you stand to face
(a) east and turn clockwise to face north?
(b) south and turn clockwise to face east?
(c) west and turn clockwise to face east?
Solution :
(a) 34 of a revolution.
(b) 34 of a revolution.
(c) 12 of a revolution.

Question 5.
Find the number of right angles turned through by the hour hand of a clock when it goes from
(a) 3 to 6
(b) 2 to 8
(c) 5 to 11
(d) 10 to 1
(e) 12 to 9
(f) 12 to 9
Solution :
(a) 1 right angle
(b) 2 right angles
(c) 2 right angles
(d) 1 right angle
(e) 3 right angles
(f) 2 right angle

Question 6.
Now many right angles do you make if you start facing
(a) south and turn clockwise to the west?
(b) north and turn anti-clockwise to the east?
(c) west and turn to west?
(d) south and turn to the north?
Solution :
(a) 1 right angle
(b) 3 right angles
(c) 4 right angles
(d) 2 right angles
(clockwise or anticlockwise).

Question 7.
Where will the hour’s hand of a clock stop if it starts
(a) from 6 and turns through 1 right angle?
(b) from 8 and turns through 2 right angles?
(c) from 10 and turns through 3 right angles?
(d) from 7and turns through 2 straight angles?
Solution :
(a) 9
(b) 2
(c) 7
(d) 7.
(We should consider only clockwise direction here).

Exercise 5.3

Question 1.
Match the following :
(i) Straight angle (a) Less than one-fourth a revolution.
(ii) Right angle    (b) More than half a revolution
(iii) Acute angle   (c) Half of a revolution
(iv) Obtuse angle (d) One-fourth a revolution
(v) Reflex angle    (e) Between 14 and 12 of a revolution (f) One complete revolution.
Solution :
(i) Straight angle ↔ (c) Half of a revolution
(ii) Right angle ↔ (d) One-fourth a revolution
(iii) Acute angle ↔ (a) Less than one-fourth a revolution
(iv) Obtuse angle ↔ (e) Between 1/4 and 1/2 of a revolution
(v) Reflex angle ↔ (b) More than half a revolution.

Question 2.
Classify each one of the following angles as right, straight, acute, obtuse or reflex :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 10
Solution :
(a) Acute
(b) Obtuse
(c) Right
(d) Reflex
(e) Straight
(f) Acute.

Exercise 5.4

Question 1.
What is the measure of
(i) a right angle
(ii) a straight angle?
Solution :
(i) The measure of a right angle is 90°.
(ii) The measure of a straight angle is 180°.

Question 2.
Say True or False:
(a) The measure of an acute angle < 90°.
(b) The measure of an obtuse angle of < 90°.
(c) The measure of reflex angle > 180°.
(d) The measure of one complete revolution = 360°.
(e) If m∠A = 53° and m∠B = 35°, then m∠A > m∠B.
Solution :
(a) True
(b) False
(c) True
(d) True
(e) True.

Question 3.
Write down the measures of
(a) some acute angles.
(b) some obtuse angles. (give at least two examples of each).
Solution :
(a) 23°, 89°
(b) 91°, 179°.

Question 4.
Measure the angles given below, using the Protractor and write down the measure.
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 11
Solution :
(a) 45°
(b) 125°
(e) 90°
(d) ∠l = 40°, ∠2 = 125° and ∠3 = 95°.

Question 5.
Which angle has a large measure? First, estimate and then measure.
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 12
Measure of angle A =
Measure of angle B =
Solution :
Measure of angle A = 40°
Measure of angle B = 65°
The angle B has a larger measure.

Question 6.
From these two angles which have larger measure? Estimate and then confirmed by the measuring them.
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 13
Solution :
Measure of first angle = 45°
Measure of second angle = 60°
The second angle has larger measure.

Question 7.
Fill in the blanks with acute, obtuse, right or straight:
(a) An angle whose measure is less than that of a right angle is. ____________
(b) An angle whose measure is greater than that of a right angle is. ____________
(c) An angle whose measure is the sum of the measures of two right angles is. ____________
(d) When the sum of the measures of two angles is that of a right angle, then each one of them is. ____________
(e) When the sum of the measures of two angles is that of a straight angle and if one of them is acute than the other should be ____________.
Solution :
(a) An angle whose measure is less than that of a right angle is an acute angle.
(b) An angle whose measure is greater than that of a right angle is the obtuse angle (if the angle is less than 180°)
(c) An angle whose measure is the sum of the measures of two right angles is the straight angle.
(d) When the sum of the measures of two angles is that of a right angle, then each one of them is acute.
(e) When the sum of the measures of two angles is that of a straight angle and if one of them is acute than the other should be an obtuse angle.

Question 8.
Find the measure of the angle shown in each figure. (First, estimate with your eye, and then find the actual measurements with a protractor).
(a)
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 14

(b)
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 15

(c)
tiwari academy class 6 maths Chapter 5 Understanding Elementary Shapes 16

(d)
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 17

Solution :
Estimation with eye
(a) 45°
(b) 125°
(c) 60°
(d) 135° actual measure with a protractor
Solution :
(a) 40°
(b) 130°
(c) 65°
(d) 135°

Question 9.
Find the angle measure between the hands of the clock in each figure :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 18
Solution :

Question 10.
Investigate :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 19
In the given figure, the angle measures 30°. Look at the same figure through a magnifying glass. Does the angle become larger? Does the size of the angle change?
Solution :
No!; No!

Question 11.
Measure and classify each angle :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 20

Angle Measure Type
∠AOC
∠AOB
∠BOC
∠DOC
∠DOA
∠DOB

Solution :
Angle ∠AOB ∠AOC ∠BOC ∠DOC ∠DOA ∠DOB Measure 40° 120° 80° 105° 140° 180° Type acute obtuse acute obtuse straight.

Exercise 5.5

Question 1.
Which of the following are models for perpendicular lines :
(a) The adjacent edges of a table top.
(b) The lines of a railway track.
(c) The line segments forming the letter “L”.
(d) The letter V.
Solution :
(a) and (c) are models for perpendicular lines.

Question 2.
Let PQ¯¯¯¯¯¯¯¯ be the perpendicular to the line segment XY¯¯¯¯¯¯¯¯ . Let PQ and XY¯¯¯¯¯¯¯¯ intersect in the point A. What is the measure of ∠PAY ?
Solution :
The measure of ∠PAY is 90°.

Question 3.
There are two “set-squares ” in your box. What are the measures of the angles that are formed at their comers? Do they have any angle measure that is common?
Solution :
One is a 30° – 60° – 90° set square; the other is a 45° – 45° – 90° set square. The angle of measure 90° (i.e., a right angle) is common between them.

Question 4.
Study the diagram. The line l is perpendicular to line m.
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 21
(a) Is CE = EG?
(b) Does PE bisect CG?
(c) Identify any two line segments for which PE is the perpendicular bisector.
(d) Are these true?
(i) AC > EG
(ii) CD = GH
(iii) BC < EH.
Solution :
(a) CE= CD + DE
= 1 + 1=2 units
EG = EF + FG
= 1 + 1=2 units
∴ CE= EG

(b)
∴ CE = EG
∴ E is the mid-point of CG.
∴ Line PE bisects line segment CG

(c)
∴ DE = EF = 1 unit
∴ PE is the perpendicular bisector of line segment DF
∴ BE = EH = 3 units
∴ PE is the perpendicular bisector of BH

(d)
(i)
true
(ii) true
(iii) true

 

Exercise 5.6

Question 1.
Name the types of following triangles:
(a) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.
(b) ΔABC with AB = 8.7cm, AC =7cm andBC = 6 cm.
(c) ΔPQR such that PQ = QR = PR = 5 cm.
(d) ΔDEF with m ∠D = 90°.
(e) ΔXYZ with m ∠Y = 90° and XY = YZ.
(f) ΔLMN with m ∠L = 30°, m ∠M =70° and m ∠N = 80°.
Solution :
(a) Scalene triangle
(b) Scalene triangle
(c) Equilateral triangle
(d) Right angled triangle
(e) Isosceles right triangle
(f) Acute angled triangle.

Question 2.
Match the following :

Measures of triangle Type triangle
(i) 3 sides of equal length (a) Scalene
(ii) 2 sides of equal length (b) Isosceles right angled
(iii) All sides are of different length (c) Obtuse angled
(iv) 3 acute angles (d) Right angled
(v) 1 right angle (e) Equilateral
(vi) 1 obtuse angle (f) Acute-angled
(vii) 1 right angle with two sides of equal length (g) Isosceles

Solution :

Measures of triangle Type triangle
(i) 3 sides of equal length (e) Equilateral
(ii) 2 sides of equal length (g) Isosceles
(iii) All sides are of different length (a) Scalene
(iv) 3 acute angles (f) Acute-angled
(v) 1 right angle (d) Right angled
(vi) 1 obtuse angle (c) Obtuse angled
(vii) 1 right angle with two sides of equal length (b) Isosceles right angled

Question 3.
Name each of the following triangles in two different ways : (you may judge the nature of the angle by observation).
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 22
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 23
Solution :
(a) Acute-angled and isosceles.
(b) Right-angled and scalene.
(c) Obtuse-angled and isosceles.
(d) Right-angled and isosceles.
(e) Equilateral and acute-angled.
(f) Obtuse-angled and scalene.

Question 4.
Try to construct triangles using matchsticks. Some are shown here. Can you make a triangle with
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 24
(a) 3 matchsticks?
(b) 4 matchsticks?
(c) 5 matchsticks?
(d) 6 matchsticks?
(Remember you have to use all the available matchsticks in each case). Name the type of triangle in each case. If you cannot make a triangle, think of reasons for it.
Solution :
(a) Yes, an equilateral triangle.
(b) No, since the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
(c) Yes, an isosceles triangle.
(d) Yes, an equilateral triangle.

Exercise 5.7

Question 1.
Say True or False :
(a) Each angle of a rectangle is a right angle.
(b) The opposite sides of a rectangle are equal in length.
(c) The diagonals of a square are perpendicular to one another.
(d) All the sides of a rhombus are of equal length.
(e) All the sides of a parallelogram are of equal length.
(f) The opposite sides of a trapezium are parallel.
Solution :
(a) True
(b) True
(c) True
(d) True
(e) False
(f) False.

Question 2.
Give reasons for the following :
(a) A square can be thought of as a special rectangle.
(b) A rectangle can be thought of as a special parallelogram.
(c) A square can be thought of as a special rhombus.
(d) Squares, rectangles, parallelograms are all quadrilaterals.
(e) Square is also a parallelogram.
Solution :
(a) A rectangle with all sides equal becomes a square.
(b) A parallelogram with each angle a right angle becomes a rectangle.
(c) A rhombus with each angle a right angle becomes a square.
(d) All these are four-sided polygons made of line segments.
(e) The opposite sides of a square are parallels, so it is a parallelogram.

Question 3.
A figure is said to be regular, if its sides are equal in length and angles are equal in measure. Can you identify the regular quadrilateral?
Solution :
A square is a regular quadrilateral.

Exercise 5.8

Question 1.
Examine whether the following are polygons. If anyone among them is not, say why?
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 25
Solution :
(a) is not a closed figure and hence is not a polygon.
(b) is a polygon of six sides.
(c) is not a polygon since it is not made of line segments.
(d) is not a polygon since it is not made of line segments.

Question 2.
Name each polygon:
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 26

Make two more examples of each of these.
Solution :
(a) A Quadrilateral
(b) A Triangle
(c) A Pentagon (5-sided)
(d) An Octagon (8-sided).
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 27

Question 3.
Draw a rough sketch of a regular hexagon. Connecting any three of its vertices, draw a triangle. Identify the type of the triangle you have drawn.
Solution :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 28
The triangle ABC drawn is an obtuse-angled isosceles triangle.

Question 4.
Draw a rough sketch of a regular octagon. (Use squared paper if you wish). Draw a rectangle by joining exactly four of the vertices of the octagon.
Solution :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 29
Question 5.
A diagonal is a line segment that joins any two vertices of the polygon and is not a side of the polygon. Draw a rough sketch of a pentagon and draw its diagonals.
Solution :
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 30

Exercise 5.9

Question 1.
Match the following :
(a) Cone
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 31
(b) Sphere
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 32
(c)
Cylinder
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 33
(d)
Cuboid
NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes 34
(e)
Pyramid
tiwari academy class 6 maths Chapter 5 Understanding Elementary Shapes 35
Give two new examples of each shape.
Solution :
(a) ↔ (ii)
(b) ↔ (iv)
(c) ↔ (v)
(d) ↔ (iii)
(e) ↔ (i).
(i) Birthday cap, ice-cream cone
(ii) Tennis Ball, laddu
(iii) Road-roller, gas cylinder
(iv) Brick, book
(v) Pyramids of Egypt, right pyramid.

Question 2.
What shape is
(a) Your installments box?
(b) A brick?
(c) A matchbox?
(d) A road-roller?
(e) A sweet laddu?
Solution :
(a) Cuboid
(b) Cuboid
(c) Cuboid
(d) Cylinder
(e) Sphere.

In-Text Questions