Class 6 - Maths - Understanding Elementary Shapes

Exercise 5.1

Question 1:

What is the disadvantage in comparing line segments by mere observation?

There may be chance of error due to improper viewing.

Question 2:

Why is it better to use a divider than a ruler, while measuring the length of a line segment?

It is better to use a divider than a ruler, because the thickness of the ruler may cause

difficulties in reading off her length. However divider gives up accurate measurement.

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.]

AB = 6.5 cm, AC = 3cm, CB = 3.5 cm

AC + CB = 3 cm + 3.5 cm = 6.5 cm = AB

Yes. AB = AC + CB

Question 4:

If A, B, C are three points on a line such that AB = 5 cm, BC = 3cm and AC = 8 cm, which one of them lies between the other two?

AC is the longest line segment, thus B is the point between A and C.

Question 5:

Verify whether D is the mid-point of AG.

From the figure, AD = 3 units, DG = 3 units.

Thus, D is the mid-point of AG.

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?

B is the mid-point of AC.

=> AB = BC ...........1

And C is the mid-point of BD.

=> BC = CD ..........2

From equation 1 and 2, we get

AB = CD

Question 7:

Draw five triangles and measure their sides. Check in each case, of the sum of the lengths of any two sides is always less than the third side.

Yes, the sum of two sides of a triangle 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

We may observe that in 1 complete clockwise rotation, the hour hand will rotate by 3600

(a) When the hour hand goes from 3 to 9 clockwise, it will rotate by 2 right angles or 1800

So, Fraction = 180/360 = 1/2

(b) When the hour hand goes from 4 to 7 clockwise, it will rotate by 1 right angle or 900

So, Fraction = 90/360 = 1/4

(c) When the hour hand goes from 7 to 10 clockwise, it will rotate by 1 right angle or 900

So, Fraction = 90/360 = 1/4

(d) When the hour hand goes from 12 to 9 clockwise, it will rotate by 3 right angles or 2700

So, Fraction = 270/360 = 3/4

(e) When the hour hand goes from 1 to 10 clockwise, it will rotate 3 right angles or 2700

So, Fraction = 270/360 = 3/4

(f) When the hour hand goes from 6 to 3 clockwise, it will rotate 3 right angles or 2700

So, Fraction = 270/360 = 3/4

Question 2:

Where will the hand of a clock stop if it:

(a) starts at 12 and make 1/2 of a revolution, clockwise?

(b) starts at 2 and makes 1/2 of a revolution, clockwise?

(c) starts at 5 and makes 1/2 of a revolution, clockwise?

(d) starts at 5 and makes 3/4 of a revolution, clockwise?

In 1 complete clockwise revolution, the hand of a clock will rotate by 3600

(a) If the hand of a clock starts at 12 and makes 1/2 of a revolution clockwise, then it will

rotate by 180 degree and hence, it will stop at 6.

(b) If the hand of a clock starts at 2 and makes 1/2 of a revolution clockwise, then it will

rotate by 180 degree and hence, it will stop at 8.

(c) If the hand of a clock starts at 5 and makes 1/4 of a revolution clockwise, then it will

rotate by 90 degree and hence, it will stop at 8.

(d) If the hand of a clock starts at 5 and makes 3/4 of a revolution clockwise, then it will

rotate by 270 degree and hence, it will stop 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  of a revolution clockwise?

(c) West and makes 3/4 of a revolution, clockwise?

(d) South and make one full revolution?

(Should we specify clockwise or anti-clockwise for this last question? Why not?)

If we revolve one complete round in either clockwise or anti-clockwise direction, then we will

revolve by 360 degree and the two adjacent directions will be at 90 degree or 1/4 of a

complete revolution away from each other.

(a) If we start facing East and make 1/2 of a revolution clockwise, then we will face the West direction.

(b) If we start facing East and make 1  of a revolution clockwise, then we will face the West direction.

(c) If we start facing East and make 3/4 of a revolution clockwise, then we will face the West direction.

(d) If we start facing South and make a full revolution, then we will again face the South direction.

In case of revolving by 1 complete round, the direction in which we are revolving does not

matter. In both cases, clockwise or anti clockwise, we will back at our initial position.

Question 4:

What part of a revolution have you turned through if you stand facing:

(a) East and turn clockwise to face North?

(b) South and turn clockwise to face East?

(c) West and turn clockwise to face East?

If we revolve one complete round in either clockwise or anti-clockwise direction, then we will

revolve by 360 degree and the two adjacent directions will be at 90 degree or 1/4 of a

complete revolution away from each other.

(a) If we start facing East and turn clockwise to face North, then we have to make 3/4 of a revolution.

(b) If we start facing West and turn clockwise to face East, then we have to make 1/2 of a revolution.

(c) If we start facing South and turn clockwise to face East, then we have to make 3/4 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 6

The hour hand of clock revolves by 360 degree or 4 right angles in 1 complete round.

(a) The hour hand of a clock revolves by 90 degree of 1 right angle when it goes from 3 to 6.

(b) The hour hand of a clock revolves by 180 degree of 2 right angles when it goes from 2 to 8.

(c) The hour hand of a clock revolves by 180 degree of 2 right angles when it goes from 5 to 11.

(d) The hour hand of a clock revolves by 90 degree of 1 right angle when it goes from 10 to 1.

(e) The hour hand of a clock revolves by 270 degree of 3 right angles when it goes from 12 to

9.

(f) The hour hand of a clock revolves by 180 degree of 2 right angles when it goes from 12 to 6.

Question 6:

How many right angles do you make if you start facing:

(a) South and turn clockwise to west?

(b) North and turn anti-clockwise to east?

(c) West and turn to west?

(d) South and turn to north?

If we revolve one complete round in either clockwise or anti clockwise direction, then we will

revolve by 360 degree or 4 right angles and the two adjacent directions will be at 90 degree or

1 right angle away from each other.

(a) If we start facing South and turn clockwise to West, then we make 1 right angle.

(b) If we start facing North and turn anti clockwise to East, then we make 2 right angles.

(c) If we start facing West and turn to West, then we make 1 complete round or 4 right angles.

(d) If we start facing South and turn to West, then we make 2 right angles.

Question 7:

Where will the hour 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 7 and turns through 2 straight angles?

In 1 complete revolution (clockwise or anti clockwise), the hour hand of a clock will rotate by

360 degree or 4 right angles.

(a) If the hour hand of a clock starts from 6 and turns through 1 right angle, then it will stop at

9.

(b) If the hour hand of a clock starts from 8 and turns through 3 right angles, then it will stop at

2.

(c) If the hour hand of a clock starts from 10 and turns through 3 right angles, then it will stop

at 7.

(d) If the hour hand of a clock starts from 7 and turns through 2 straight angles, then it will

stop at 7.

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 1/4 and 1/2 of a revolution

(f) one complete revolution

(i) Straight angle is of 180 degree and half of a revolution is 180 degree

Hence, (i)    -->   (c)

(ii) Right angles are of 90 degree and one-fourth of a revolution is 90 degree

Hence, (ii)   -->   (d)

(iii) Acute angles are angles less than 90 degree. Also, less than one-fourth of a revolution is

the angle less than 90 degree.

Hence, (iii)  -->   (a)

(iv) Obtuse angles are the angles greater than 90 degree but less than 180 degree. Also,

between 1/4 and 1/2 of a revolution is the angle whose measure lies between 90 and 180

degree.

Hence, (iv)   -->   (e)

(v) Reflex angles are the angles greater than 180 degree but less than 360 degree. Also, more

than half a revolution is the angle whose measure is greater than 180 degree.

Hence, (v)    -->   (b)

Question 2:

Classify each one of the following angles as right, straight, acute, obtuse or reflex:

(a) Acute angle as its measure is less than 90 degree.

(b) Obtuse angle as its measure is more than 90 degree but less than 180 degree.

(c) Right angle as its measure is 90 degree.

(d) Reflex angle as its measure is more than 180 degree but less than 360 degree.

(e) Straight angle as its measure is 180 degree.

(f) Acute angle as its measure is less than 90 degree.

Exercise 5.4

Question 1:

What is the measure of (i) a right angle? (ii) a straight angle?

(i) The measure of a right angle is 90 degree.

(ii) The measure of a right angle is 180 degree.

Question 2:

Say True or False:

(a) The measure of an acute angle < 90.

(b) The measure of an obtuse angle < 90.

(c) The measure of a reflex angle > 180.

(d) The measure of on complete revolution = 360.

(e) If mÐA = 53 degree and mÐ B = 35 degree, then mÐ A > mÐ B.

(a) True

The measure of an acute angle is less than 90 degree.

(b) False

The measure of an obtuse angle is greater than 90 degree but less than 180 degree.

(c) True

The measure of a reflex angle is greater than 180 degree.

(d) True

The measure of one complete revolution is 360 degree.

(e) True

If mÐA = 53 degree and mÐ B = 35 degree, then mÐ A > mÐ B

Question 3:

Write down the measure of:

(a) some acute angles (b) some obtuse angles

( give at least two examples of each)

(a) An acute angle is less than 90 degree.

Example: 300, 450, 600, etc.

(b) The measure of an obtuse angle is greater than 90 degree but less than 180 degree.

Example: 1200, 1350, 1600, etc.

Question 4:

Measure the angles given below, using the protractor and write down the measure:

By using the protractor, we find that

(a) 400            (b) 1300               (c) 900            (d) 600

Question 5:

Which angle has a large measure? First estimate and then measure:

Measure of angle A =

Measure of angle B =

Measure of angle A = 40 degree

Measure of angle B = 60 degree

So, angle B has the greater measure than angle A.

Question 6:

From these two angles which has larger measure? Estimate and then confirm by measuring them:

The measures of these angles are 45 degree and 55 degree. Therefore, the angle shown in 2nd

figure is greater.

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 then the other should be ________.

(a) acute angle             (b) obtuse angle               (c) straight angle             (d) acute angle

(e) obtuse angle

Question 8:

Find the measure of the angle shown in each figure. (First estimate with your eyes and then find the actual measure with a protractor).

The measures of the angles shown in the above figure are 400, 1300, 650 and 1350 respectively.

Question 9:

Find the angle measure between the hands of the clock in each figure:

(a) 900             (b) 300            (c) 1800

Question 10:

Investigate:

In the given figure, the angle measure 30 degree. Look at the same figure through a magnifying glass. Does the angle become larger?

Does the size of the angle change?

No, the measure of angle will be same.

Question 11:

Measure and classify each angle:

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’.

(a) The adjacent edges of a table top are perpendicular to each other.

(b) The lines of a railway track are parallel to each other.

(c) The line segments forming the letter ‘L’ are perpendicular to each other.

(d) The sides of letter ‘V’ are inclined at some acute angle on each other.

Hence, (a) and (c) are the 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 ÐPAY?

From the figure, it can be easily observed that the measure of ÐPAY is 90 degree.

Question 3:

There are two “set-squares” in your box. What are the measures of the angles that are formed at their corners?

Do they have any angle measure that is common?

One set-square has 450, 900, 450 and other set-square has 600, 900, 300.

They have 900 as common angle.

Question 4:

Study the diagram. The line l is perpendicular to line m.

(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 > FG (ii) CD = GH (iii) BC < EH

(a) Yes, both measure 2 units.

(b) Yes. PE bisects CG because CE = EG

(c) DF and BH

(d) (i) True. As length of AC and FG are of 2 units and 1 unit respectively.

(ii) True. As both have I unit length.

(iii) True. As the length of BC and EH are of 1 unit and 3 units respectively.

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) Δ ABC with AB = 8.7 cm, AC = 7 cm and BC = 6 cm.

(c) Δ PQR such that PQ = QR = PR = 5 cm.

(d) Δ DEF with mÐ D = 900

(e) Δ XYZ with mÐ Y = 900 and XY = YZ

(f) Δ LMN with mÐ L = 300, mÐM = 700 and mÐ N = 800.

(a) Scalene triangle

(b) Scalene triangle

(c) Equilateral triangle

(d) Right-angled triangle

(e) Isosceles right-angled triangle

(f) Acute-angled triangle

Question 2:

Match the following:

Measure of Triangle Types of Triangle

(i) 3 sides of equal length                     (a) Scalene

(ii) 2 sides of equal length                    (b) Isosceles right angle

(iii) All sides are of different length    (c) Obtuse angle

(iv) 3 acute angles                                  (d) Right angle

(v) 1 right angle                                   (e) Equilateral

(vi) 1 obtuse angle                              (f) Acute angle

(vii) 1 right angle with two sides      (g) Isosceles of equal length

(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 angle

(vii) 1 right angle with two sides         (b) Isosceles right-angled

Question 3:

Name each of the following triangles in two different ways: (You may judge the nature of angle by observation)

(a) Acute angled triangle and Isosceles triangle

(b) Right-angled triangle and scalene triangle

(c) Obtuse-angled triangle and Isosceles triangle

(d) Right-angled triangle and Isosceles triangle

(e) Equilateral triangle and acute angled triangle

(f) Obtuse-angled triangle and scalene triangle

Question 4:

Try to construct triangles using match sticks. Some are shown here.

Can you make a triangle with:

(a) 3 matchsticks?

(b) 4 matchsticks?

(c) 5 matchsticks?

(d) 6 matchsticks?

(Remember you have to use all the available matchsticks in each case)

If you cannot make a triangle, think of reasons for it.

(a) 3 matchsticks

This is an acute angle triangle and it is possible with 3 matchsticks to make a triangle because

sum of two sides is greater than third side.

(b) 4 matchsticks

This is a square, hence with four matchsticks we cannot make triangle.

(c) 5 matchsticks

This is an acute angle triangle and it is possible to make triangle with five matchsticks, in this

case sum of two sides is greater than third side.

(d) 6 matchsticks

This is an acute angle triangle and it is possible to make a triangle with the help of 6

matchsticks because sum of two sides is greater than third side.

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.

(a) True.

Each angle of a rectangle is a right angle.

(b) True.

The opposite sides of a rectangle are equal in length.

(c) True.

The diagonals of a square are perpendicular to one another.

(d) True.

All the sides of a rhombus are of equal length.

(e) False.

All the sides of a parallelogram are not of equal length.

(f) False.

The opposite sides of a trapezium are not parallel.

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 quadrilateral.

(e) Square is also a parallelogram.

(a) Because its all angles are right angle and opposite sides are equal.

(b) Because its opposite sides are equal and parallel.

(c) Because its four sides are equal and diagonals are perpendicular to each other.

(d) Because all of them have four sides.

(e) Because its opposite sides are equal and parallel.

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?

In a square, all the interior angles are of 900 and all the sides are of the same length.

Therefore, a square is a regular quadrilateral.

Exercise 5.8

Question 1:

Examine whether the following are polygons. If anyone among these is not, say why?

(a) As it is not a closed figure, therefore, it is not a polygon.

(b) It is a polygon because it is closed by line segments.

(c) It is not a polygon because it is not made by line segments.

(d) It is not a polygon because it not made only by line segments, it has curved surface also.

Question 2:

Name each polygon:

Make two more examples of each of these.

(a) The given figure is a quadrilateral as this closed figure is made of 4 line segments. Two more examples are:

(b) The given figure is a triangle as this closed figure is made of 3 line segments. Two more examples are:

(c) The given figure is a pentagon as this closed figure is made of 5 line segments. Two more examples are:

(d) The given figure is a octagon as this closed figure is made of 8 line segments. Two more examples are:

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.

ABCDEF is a regular hexagon and triangle thus formed by joining AEF is an isosceles triangle.

Question 4:

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.

ABCDEFGH is a regular octagon and CDGH is a rectangle.

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.

ABCDE is the required pentagon and its diagonals are AD, AC, BE and BD.

Exercise 5.9

Question 1:

Match the following:

(a) Cone

(b) Sphere

(c) Cylinder

(d) Cuboid

(e) Pyramid

Give two example of each shape.

(a) Cone

(b) Sphere

(c) Cylinder

(d) Cuboid

(e) Pyramid

Question 2:

What shape is:

(a) Your instrument box?            (b) A brick?                       (c) A match box?

(a) An instrument box has shape of cuboid.

(b) A brick has shape of cuboid.

(c) A match box has shape of cuboid.

(d) A road-roller has shape of cylinder.

(e) A sweet laddu has shape of Sphere.