## NCERT Solutions Class 6 Maths Chapter - 5: Understanding Elementary Shapes## Exercise 5.1
## 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.
BC = 2.3 cm AB = 4.3 cm Yes, AB = AC + CB AB = 2 cm + 2.3 cm AB = 4.3 cm It is because point C lies between the line segments AB.
AC is a bigger line segment. Thus, AC = AB + BC It shows that point B lies between point A and C. AC = AB + BC AC = 5 cm + 3 cm AC = 8 cm
There are total seven points in the above line segment. Mid-point = 7 + 1/2 = 4 Thus, 4th point (D) is the mid-point of the line segment AG.
Let's assume the above line segment as: B is the midpoint of AC. A mid-point equally divides a line into two equal parts. Thus, AB = BC C is the mid-point of BD Thus, BC = CD Hence, we can say that AB = BC = CD AB = CD
The five triangles are: Sum of length of two sides of the first triangle: 1 cm + 1 cm = 2cm Length of the third side = 1 cm Sum of length of two sides of the second triangle: 2 cm + 2 cm = 4 cm Length of the third side = 2 cm Sum of length of two sides of the third triangle: 3 cm + 3 cm = 6cm Length of the third side = 3 cm Sum of length of two sides of the fourth triangle: 4 cm + 4 cm = 8cm Length of the third side = 4 cm Sum of length of two sides of the fifth triangle: 5 cm + 5 cm = 10 cm Length of the third side = 5 cm Thus, sum of length of two sides of a triangle is never less than the length of its third side. ## Exercise 5.2
3 to 9 = 6 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 6/12 = 1/2
4 to 7 = 3 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 3/12 = 1/4
7 to 10 = 3 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 3/12 = 1/4
12 to 9 = 9 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 9/12 = 3/4
1 to 10 = 9 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 9/12 = 3/4
6 to 3 = 9 hours There are total 12 hours in a watch. Fraction of clockwise revolution = Number of hours a clock turn/Total number of hours = 9/12 = 3/4
½ of a revolution = 12/2 = 6 hours Thus, starting at 12 and making a ½ of a revolution clockwise is equal to 6.
½ of a revolution = 12/2 = 6 hours Thus, starting at 2 and making a ½ of a revolution clockwise is equal to 8.
¼ of a revolution = 12/4 = 3 hours Thus, starting at 5 and making a ¼ of a revolution clockwise is equal to 8.
¾ of a revolution = (12 � 3)/4 = 9 hours Thus, starting at 5 and making a ¾ of a revolution clockwise is equal to 2.
1 revolution is equal to full turn of 360 degrees.
½ revolution = 180 degrees or 6 hours Turning east and making ½ of a revolution clockwise, result in the west direction.
3/2 x 12 hours = 18 hours Or 3/2 x 360 = 540 degrees Thus, the clock will make one full turn + one half turn. = East to East and again from East to West
Or 270 degrees 9 hours in the anti-clockwise direction will turn it to the Similarly, the same 9 hours in the clockwise direction will turn it to the South direction.
(Should we specify clockwise or anti-clockwise for this last question? Why not? )
No, we are not required to specify clockwise or anti-clockwise for this last question because the clock will turn at the
(a) East and turn clockwise to face north?
(b) South and turn clockwise to face east?
(c) West and turn clockwise to face east?
(a) 3 to 6
3 hours covered by a hour hand = 1 right angle Only one right angle is covered by the hour hand of a clock when it goes from 3 to 6. (b) 2 to 8
3 hours covered by a hour hand = 1 right angle 6 hours = 2 right angles Two right angles are covered by the hour hand of a clock when it goes from 2 to 8. (c) 5 to 11
3 hours covered by a hour hand = 1 right angle 6 hours = 2 right angles Two right angles are covered by the hour hand of a clock when it goes from 5 to 11. (d) 10 to 1
3 hours covered by a hour hand = 1 right angle Only one right angle is covered by the hour hand of a clock when it goes from 10 to 1. (e) 12 to 9
3 hours covered by a hour hand = 1 right angle 9 hours = 3 right angles Three right angles are covered by the hour hand of a clock when it goes from 12 to 9. (f) 12 to 6
3 hours covered by a hour hand = 1 right angle 6 hours = 2 right angles Two right angles are covered by the hour hand of a clock when it goes from 12 to 6.
(a) South and turn clockwise to west?
One turn = 1 right angle (b) North and turn anti-clockwise to east?
One turn = 1 right angle Three turns = Three right angles (c) West and turn to west?
(d) South and turn to north?
(a) From 6 and turns through 1 right angle?
6 + 3 = 9 Thus, the hour hand of a clock stops at 9 if it starts from 6 and turns through 1 right angle. (b) From 8 and turns through 2 right angles?
2 right angles = 6 hours 8 + 6 = 2 (in the clock) Thus, the hour hand of a clock stops at 2 if it starts from 8 and turns through 2 right angles. (c) From 10 and turns through 3 right angles?
3 right angles = 9 hours 10 + 9 = 7 (in the clock) Thus, the hour hand of a clock stops at 7 if it starts from 10 and turns through 3 right angles. (d) From 7 and turns through 2 straight angles?
There are 2 straight angles in a clock. We can also say that 1 straight angle is equal to 180 degrees, where 1 revolution is of 360 degrees. 2 straight angles = 12 hours Thus, the hour hand of a clock stops at the same time 7 if it starts from 7 and turns through 2 straight angles. ## Exercise 5.3
(i) Straight angle (c) Half of a revolution (ii) Right angle (d) One-fourth of a revolution (iii) Acute angle (a) less than one-fourth of a revolution (iv) Obtuse angle (e) between 1/4 and 1/2 of a revolution (v) Reflex angle (b) More than half a revolution
One revolution = 360 degrees Half revolution = 180 degrees 1/4th of a revolution = 90 degrees
(a)
(b)
(c)
(d)
(e)
(f)
## Exercise 5.4
(i) A right angle?
(ii) A straight angle?
(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 one complete revolution = 360°.
(e) If m∠ A = 53° and m∠ B = 35°, then m∠ A > m∠ B.
3. Write down the measures of (a) Some acute angles.
(b) Some obtuse angles.
(Give at least two examples of each).
- 45°
- 120°
- 90°
- 60°, 130°, 90°
Measure of Angle A = 40° Measure of Angle B = 68°
Angle B has the larger measure.
(a) An angle whose measure is less than that of a right angle is
1 right angle = 90 degrees (b) An angle whose measure is greater than that of a right angle is
1 right angle = 90 degrees (c) An angle whose measure is the sum of the measures of two right angles is
1 right angle = 90 degrees 2 right angles = 180 degrees A straight angle is the angle formed on a straight line, whose value is equal to 180 degrees. (d) When the sum of the measures of two angles is that of a right angle, then each one of them is
1 right angle = 90 degrees A + B = 90 Thus, both two angles A and B have value less than 90 degrees. For example, 35 + 55 = 90 (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 + B = 180 Straight angle = 180 If A is acute, then B should be obtuse to obtain the sum of 180. For example, A = 75 B = 180 - 75 B = 105 degrees
There are 12 hands of a clock. 12 hands = 360 degree or 1 revolution 1 hand = 360/12 = 30 degrees Thus, each hand of a clock makes an angle of 30 degrees. - 9.00 a.m.
**Answer**: 90 degrees The difference of 3 digits makes it as an angle of 90 degrees. - 1.00 p.m.
**Answer**: 30 degrees The difference of 1 digit makes it as an angle of 30 degrees. - 6.00 p.m.
**Answer**: 180 degrees The difference of 6 digits makes it as a straight angle of 180 degrees.
No. The angle does not become large while watching through a magnifying class. It has the same value.
The angle with value less than 90 degrees is classified as an acute angle. The angle with value greater than 90 degrees is classified as an obtuse angle. The angle with value equal to 180 degrees is classified as a straight angle. ## Exercise 5.5
(a) The adjacent edges of a table top.
The adjacent edges of a table top are perpendicular to each other, i.e., at 90 degrees. (b) The lines of a railway track.
The lines of a railway track are parallel to each other, not perpendicular. (c) The line segments forming the letter 'L'.
The line segments forming the letter 'L' are perpendicular to each other, i.e., at 90 degrees. (d) The letter V.
The letter V does not form an angle of 90 degrees. Hence, they are not perpendicular.
The measure of ∠PAY is equal to 90 degrees.
Yes. Both set squares have one angle in common, i.e., 90 degrees.
The gap between the two lines is equal, i.e. 2 units. Hence, CE = EG.
Bisect means to intersect a line. PE intersects the line CG. Hence, the answer is yes.
There are multiple line segments for which PE is the perpendicular bisector. - AH
- AF
- AG
- BF
- BG
- BH
- CF
- CG
- CH
- DF
- DG
- DH
We can write any of the two line segments in the answer.
(i) AC > FG
Difference of AC = 3 - 1 = 2 units Difference of FG = 7 - 6 = 1 unit Hence, AC > FG (ii) CD = GH
Difference of CD = 4 - 3 = 1 unit Difference of GH = 8 - 7 = 1 unit Hence, CD = GH (iii) BC < EH.
Difference of BC = 3 - 2 = 1 unit Difference of EH = 8 -5 = 3 units Hence, BC < EH Thus, all of the above answers are true. ## Exercise 5.61. Name the types of following triangles: (a) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.
A triangle with all the three sides of unequal length is called a scalene triangle. (b) ∆ABC with AB = 8.7 cm, AC = 7 cm and BC = 6 cm.
A triangle with all the three sides of unequal length is called a scalene triangle. (c) ∆PQR such that PQ = QR = PR = 5 cm.
A triangle with all the three sides of equal length is called an equilateral triangle. (d) ∆DEF with m∠D = 90°
A triangle with any one angle as a right angle is called a right angle triangle. (e) ∆XYZ with m∠ Y = 90° and XY = YZ.
A triangle with two equal sides and one right angle is called an isosceles right angled triangle. (f) ∆LMN with m∠ L = 30°, m∠ M = 70° and m∠ N = 80°.
A triangle with angles less than 90 degrees is called an acute angle triangle.
Measures of Triangle Type of 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 (v) 1 right angle (vi) 1 obtuse angle (vii) 1 right angle with two sides of equal length
4. Try to construct triangles using match sticks. Some are shown here. Can you make a triangle with?
It is an equilateral triangle because all the three matchsticks are of equal length.
It is an isosceles triangle because the two sides of the triangle are equal.
It is an isosceles triangle because the two sides of the triangle are equal.
It is an equilateral triangle because each side is made of two matchsticks of equal size. (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. ## Exercise 5.7
(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) 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.
A square is a regular quadrilateral with equal sides and angles. ## Exercise 5.8
- No, it is not a polygon. A polygon is a closed figure.
- Yes, it a polygon because it is an enclosed figure made of line segments.
- No, it is not a polygon. A polygon is an enclosed figure made of only line segments.
- No, it is not a polygon. A polygon is an enclosed figure made of only line segments.
- Quadrilateral
- Triangle
- Pentagon
- Octagon
A four sided polygon is known as a A three sided polygon is known as a A five sided polygon is known as a An eight sided polygon is known as an
All the sides of a regular hexagon are equal. The type of triangle formed by joining any two points of a hexagon is an
The rectangle formed by joining four of its vertices is:
A pentagon is a five-sided polygon. The diagonals of a pentagon are shown below: ABCDE is a pentagon and AC, AD, EB, EC, DB are the diagonals of the pentagon. ## Exercise 5.9
(a) Cone
The examples of cone are: - Funnel
- Ice-cream cone
(b) Sphere
The examples of sphere are: - Ball
- Moon
(c) Cylinder
The examples of cylinder are: - Tube
- Rod
(d) Cuboid
The examples of cuboid are: - Table
- Book
(e) Pyramid
The examples of pyramid are: - Tent
- Temples
Give two new examples of each shape.
(a) Your instrument box?
(b) A brick?
(c) A match box?
(d) A road-roller?
(e) A sweet laddu?
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