Sunday, December 17, 2023

Class – 6 CH-5 UNDERSTANDING ELEMENTARY SHAPES MATHS NCERT SOLUTIONS

Class – 6 CH-5 UNDERSTANDING ELEMENTARY SHAPES

MATHS NCERT SOLUTIONS

Exercise 5.1

Question 1:

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

Solution 1:

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?

Solution 2:

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.]

Solution 3:

Yes.

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

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

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?

Solution 4:

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.

Solution 5:

AD = 3 units, DG = 3 units

AD = DG.

Thus, D is the mid-point.

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 6:

B is the mid-point of AC.

∴ AB = BC ... (i)

And C is the mid-point of BD .

∴ BC = CD ... (ii)

From equation (i) and (ii), 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.

Solution 7:

Yes, 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

Solution 1:

(a) or two right angles

(b) or one right angle

(d) or three right angles.

(e) or three right angles.

(f) or three right angles.

Question 2:

Where will the hand of a clock stop if it:

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

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

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

Solution 2:

(a) At 6

(b) At 8

Question 3:

Which direction will you face if you start facing:

(a) East and make of a revolution clockwise?

(b) East and make 1 of a revolution clockwise?

(d) South and make one full revolution?

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

Solution 3:

(a) West (b) West (c) North

(d) South

(For Solution (d), it is immaterial whether we turn clockwise or anticlockwise, 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 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?

Solution 4:

(a)

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

(d) 10 to 1

(e) 12 to 9

(f) 12 to 6

Solution 5:

(a) One right angle

(b) Two right angles

(e) Three right angles (f) Two right angles

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?

Solution 6:

(a) One right angle (b) Three 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?

Solution 7:

(a) At 9

(b) At 2

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 and of a revolution

(f) one complete revolution

Solution 1:

(i) 🠊 (c)

(ii) 🠊 (d)

(iii) 🠊 (a)

(iv) 🠊 (e)

(v) 🠊 (b)

Question 2:

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

Solution 2:

(a) Acute angle

(b) Obtuse angle

(d) Reflex angle

(e) Straight angle

(f) Acute angle

Exercise 5.4

Question 1:

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

Solution 1:

(i) 90 °

(ii) 180 °

Question 2:

Say True or False:

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

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

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

(e) If m∠A = 53° and m∠∠ = 35°, then mA > m∠B.

Solution 2:

(a) True

(b) False

(d) True

(e) True

Question 3:

Write down the measure of:

(a) some acute angles

(b) some obtuse angles

(give at least two examples of each)

Solution 3:

(a) 35°,20°

(b) 110° ,135°

Question 4:

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

Solution 4:

(a) 40° (b) 130°

Question 5:

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

Measure of angle A =

Measure of angle B =

Solution 5:

B has larger measure.

∠A = 40°

and ∠B = 65°

Question 6:

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

Solution 6:

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 ________________.

________________.

(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 ________________.

Solution 7:

(a) acute angle

(b) obtuse 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).

Solution 8:

(i) 30° (ii) 120 °

(iii) 60 ° (iv) 150 °

Question 9

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

9.00 a.m. 1.00 p.m. 6.00 p.m.

Solution 9:

(i) 90 (Right angle)

(ii) 30 (Acute angle)

(iii) 180 (Straight angle)

Question 10:

Investigate:

In the given figure, the angle measure 30 . Look at the same figure through a magnifying glass. Does the angle becomes larger? Does the size of the angle change?

Solution 10:

No, the measure of angle will be same.

Question 11:

Angle Measure Type

∠AOB

∠AOC

∠BOC

∠DOC

∠DOA

∠DOB

Solution 11:

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

Measure 40° 130° 90° 90° 140° 180°

Type Acute Obtuse Right Right 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.

(d) The letter V.

Solution 1:

(a) Perpendicular

(b) Not perpendicular

(d) Not perpendicular

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 2:

∠PAY = 90°

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?

Solution 3:

One set-square has 45 ,90 ,45 and other set-square has 60 ,90 ,30 . They have 90 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?

(d) Are these true? (i) AC > FG (ii) CD = GH (iii) BC < EH

Solution 4:

(a) Yes, both measure 2 units.

(b) Yes, because CE = EG

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

(d) △DEF with m∠D = 90

(e) △XYZ with m∠Y = 90 and XY = YZ

(f) △LMN with m∠L = 30, m∠M = 70 and m∠N = 80 .

Solution 1:

(a) Scalene triangle

(b) Scalene 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

Solution 2:

(i) 🠊 (e), (ii) 🠊 (g),

(iii) 🠊 (a), (iv) 🠊 (f),

(v) 🠊 (d), (vi) 🠊 (c),

(vii) 🠊 (b)

Question 3:

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

Solution 3:

(a) Acute angled triangle and Isosceles triangle

(b) Right-angled triangle and scalene 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?

(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.

Solution 4:

(a) 3 matchsticks yes, an equilateral triangle

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.

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

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 yes an equilateral triangle

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.

(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 1:

(a) True (b) 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.

(d) Squares, rectangles, parallelograms are all quadrilateral. (e) Square is also a parallelogram.

Solution 2:

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

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

(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?

Solution 3:

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?

Solution 1:

(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.

(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.

Solution 2:

(a) Quadrilateral

(b) Triangle

(d) Octagon

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 3:

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.

Solution 4:

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.

Solution 5:

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

(d) Cuboid

(e) Pyramid

Give two example of each shape.

Solution 1:

(a) Cone

(b) Sphere

(d) Cuboid

(e) Pyramid

Question 2:

What shape is:

(a) Your instrument box?

(b) A brick?

(d) A road-roller? (e) A sweet laddu?

Solution 2:

(a) Cuboid

(b) Cuboid

(d) Cylinder

(e) Sphere

Class – 6 CH-4 BASIC GEOMETRICAL IDEAS MATHS NCERT SOLUTIONS

Class – 6 CH-4 BASIC GEOMETRICAL IDEAS

MATHS NCERT SOLUTIONS

Exercise 4.1

Question 1:

Use the figure to name:

(a) Five points

(b) A line

(d) Five line segments

Solution 1:

(a) Five points are: O, B, C, D, E

(b) A line: DE, DB, OE, OB

(d) Four line segments: DE, OE, OC, OB, , OD

Question 2:

Name the line given in all possible (twelve) ways, choosing only two letters at a time from the four given.

Solution:

AB, AC, AD, BC, BD, CD, BA, CA, DA, CB, DB, DC

Question 3:

Use the figure to name:

(a) Line containing point E.

(b) Line passing through A.

Solution 3:

(a) A line containing E = AE or FE

(b) A line passing through A = AE or DE

(d) Two pairs of intersecting lines are: AD, CO andAE , FE

Question 4:

How many lines can pass though:

(a) one given point?

(b) two given points

Solution 4:

(a) Infinite number of lines can pass through one given point.

(b) Only one line can pass through two given points.

A B

Question 5:

Draw a rough figure and label suitably in each of the following cases:

(a) Point P lies on AB. (b) XY and PQ intersect at M.

Solution 5:

Question 6:

Consider the following figure of line MN. Say whether following statements are true or false in the context of the given figure:

(a) Q, M, O, N, P are points on the line MN.

(b) M, O, N are points on a line segment MN.

(d) O and N are end points of line segment OP.

(e) M is one of the end points of line segment QO .

(f) M is point on ray OP.

(g) Ray OP is different from ray OP.

(h) Ray OP same as ray OM..

(i) Ray OM. is not opposite to ray OP.

(j) O is not an initial point of NP and NM .

Solution 6:

(a) True

(b) True

(d) False

(e) False

(f) False

(g) True

(h) False

(i) False

(j) False

(k) True

EXERCISE 4.2

Question 1:

Classify the following curves as (i) Open or (ii) Closed.

Solution 1:

(a) Open curve (b) Closed curve (c) Open curve (d) Closed curve (e) Closed curve

Question 2:

Draw rough diagrams to illustrate the following:

(a) Open curve (b) Closed curve

Solution 2:

(a) Open curves: ◡◠⧽

(b) Closed curves ⃝⃞▷▭

Question 3:

Draw any polygon and shade its interior.

Solution 3:

Polygon ABCDE

Question 4:

Consider the given figure and Solution the questions:

(a) Is it a curve?

(b) Is it closed?

Solution 4:

(a) Yes, it is a curve.

(b) Yes, it is closed.

Question 5:

Illustrate, if possible, each one of the following with a rough diagram:

(a) A closed curve that is not a polygon.

(b) An open curve made up entirely of line segments.

Solution 5:

(a) ⃝

(b)

EXERCISE 4.3

Question 1:

Name the angles in the given figure:

Solution 1:

There are four angles in given figure:

∠ ABC, ∠ CDA, ∠ DAB, ∠ DCB

Question 2:

In the given diagram, name the point(s):

(a) In the interior of ∠ DOE.

(b) In the exterior of ∠EOF.

Solution 2:

(a) Point interior of ∠DOE : A

(b) Points exterior of ∠ EOF: C, A, D

Question 3:

Draw rough diagrams of two angles such that they have:

(a) One point in common.

(b) Two points in common.

(d) Four points in common.

(e) One ray in common.

Solution 3:

EXERCISE 4.4

Question 1:

Draw a rough sketch of a triangle ABC. Mark a point P in its interior and a point Q in its exterior. Is the point A in its exterior or in its interior?

Solution 1:

A is neither interior of the figure nor exterior of triangle. It is a vertex.

Question 2:

(a) Identify three triangles in the figure:

(b) Write the names of seven angles.

(d) Which two triangles have ∠B as common?

Solution 2:

(a) The three triangles are: △ABC, △ABD, △ADC

(b) Angles are: ∠ADB, ∠ ADC, ∠ ABD, ∠ ACD, ∠BAD, ∠ CAD, ∠ BAC

(d) Triangles having common ∠B: △ABC, △ABD, ⎺

Exercise 4.5

Question 1:

Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them. Is the meeting point of the diagonals in the interior or exterior of the quadrilateral?

Solution 1:

Diagonal PR and diagonal SQ meet at O, which is inside the quadrilateral.

Question 2:

Draw a rough sketch of a quadrilateral KLMN. State:

(a) Two pairs of opposite sides.

(b) Two pairs of opposite angles.

(d) Two pairs of adjacent angles.

Solution 2:

(a) Pair of opposite sides : ⎺KL and ⎺MN, ⎺KN and ⎺LM

(b) Pair of opposite angles : ∠ K and ∠ M, ∠L and ∠ N

(d) Pair of adjacent angles : ∠ K and ∠ N, ∠L and ∠ M

Question 3:

Investigate:

Use strip and fasteners to make a triangle and a quadrilateral.

Try to push inward at any one vertex of the triangle. Do the same to the quadrilateral.

Is the triangle distorted? Is the quadrilateral distorted? Is the triangle rigid?

Why is it that structures like electric towers make use of triangular shapes and not quadrilateral?

Solution 3:

O is common to both the angles  AOC and  BOC.

No, the triangle is not distorted but the quadrilateral is distorted and also the triangle is rigid.

Structures like electric towers make use of triangular shape so that they could not be distorted and they could be rigid.

Exercise 4.6

Question 1:

From the figure, identify:

(a) The centre of circle.

(b) Three radii.

(d) A chord.

(e) Two points in the interior.

(f) A point in the exterior.

(g) A sector.

(h) A segment.

Solution 1:

(a) O is the centre.

(b) Three radii: OA, OB and OC

(d) A chord: ED

(e) Interior points: O, P

(f) Exterior point: Q

(g) A sector: OAB

(h) A segment: ED

Question 2:

(a) Is every diameter of a circle also a chord?

(b) Is every chord of a circle also a diameter?

Solution 2:

(a) Yes, every diameter of a circle is also a chord.

(b) No, every chord of a circle is not a diameter.

Question 3:

Draw any circle and mark:

(a) Its centre.

(b) A radius.

(d) A sector.

Solution 3:

(a) Its centre is O.

(b) A radius is OB.

(d) A sector OAB.

Question 4:

Say true or false:

(a) Two diameters of a circle will necessarily intersect. (b) The centre of a circle is always in its interior.

Solution 4:

(a) True

(b) True