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What is the disadvantage in comparing line segments by mere observation?
By mere observation we can’t compare the line segments with slight difference in their length. We can’t say which line segment is of greater length. Hence, the chances of errors due to improper viewing are higher.
Why is it better to use a divider than a ruler, while measuring the length of a line segment?
While using a ruler, chances of error occur due to thickness of the ruler and angular viewing. Hence, using a divider accurate measurement is possible.
Draw any line segment, say . Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB?
Since given that point C lies in between A and B. Hence, all points are lying on the same line segment
. Therefore, for every situation in which point C is lying in between A and B, we may say that
AB = AC + CB
For example:
AB is a line segment of length 7 cm and C is a point between A and B such that AC = 3 cm and CB = 4 cm.
Hence, AC + CB = 7 cm
Since, AB = 7 cm
∴ AB = AC + CB is verified.
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?
Given AB = 5 cm
BC = 3 cm
AC = 8 cm
Now, it is clear that AC = AB + BC
Hence, point B lies between A and C.
Verify, whether D is the mid point of .
Since, it is clear from the figure that AD = DG = 3 units. Hence, D is the midpoint of
If B is the mid point of and C is the mid point of , where A, B, C, D lie on a straight line, say why AB = CD?
Given
B is the midpoint of AC. Hence, AB = BC (1)
C is the midpoint of BD. Hence, BC = CD (2)
From (1) and (2)
AB = CD is verified
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.
Case 1. In triangle ABC
AB= 2.5 cm
BC = 4.8 cm and
AC = 5.2 cm
AB + BC = 2.5 cm + 4.8 cm
= 7.3 cm
As 7.3 > 5.2
∴ AB + BC > AC
Hence, the sum of any two sides of a triangle is greater than the third side.
Case 2. In triangle PQR
PQ = 2 cm
QR = 2.5 cm
PR = 3.5 cm
PQ + QR = 2 cm + 2.5 cm
= 4.5 cm
As 4.5 > 3.5
∴ PQ + QR > PR
Hence, the sum of any two sides of a triangle is greater than the third side.
Case 3. In triangle XYZ
XY = 5 cm
YZ = 3 cm
ZX = 6.8 cm
XY + YZ = 5 cm + 3 cm
= 8 cm
As 8 > 6.8
∴ XY + YZ > ZX
Hence, the sum of any two sides of a triangle is greater than the third side.
Case 4. In triangle MNS
MN = 2.7 cm
NS = 4 cm
MS = 4.7 cm
MN + NS = 2.7 cm + 4 cm
6.7 cm
As 6.7 > 4.7
∴ MN + NS > MS
Hence, the sum of any two sides of a triangle is greater than the third side.
Case 5. In triangle KLM
KL = 3.5 cm
LM = 3.5 cm
KM = 3.5 cm
KL + LM = 3.5 cm + 3.5 cm
= 7 cm
As 7 cm > 3.5 cm
∴ KL + LM > KM
Hence, the sum of any two sides of a triangle is greater than the third side.
Therefore, we conclude that the sum of any two sides of a triangle is always greater than the third side.
Which direction will you face if you start facing
(a) east and make 1/2 of a revolution clockwise?
(b) east and make 1½ of a revolution clockwise?
(c) west and make 3/4 of a revolution anticlockwise?
(d) south and make one full revolution?
(Should we specify clockwise or anticlockwise for the last question? Why not?)
Revolving one complete round in a clockwise or an anticlockwise direction, we will revolve by 3600, and two adjacent directions are at 900 or 1/4 of a complete revolution away from each other.
(a) If we start facing towards the east and make 1/2 of a revolution clockwise, we will face towards the west direction.
(b) If we start facing towards the east and make 1½ of a revolution clockwise, we will face towards the west direction.
(c) If we start facing towards the west and make 3/4 of a revolution anticlockwise, we will face towards the north direction.
(d) If we start facing the south and make one full revolution, again, we will face the south direction.
In the case of revolving 1 complete revolution, either clockwise or anticlockwise, we will be back at the original position.
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 a clock revolves by 3600, or it covers 4 right angles in one complete revolution.
(a) If the hour hand of a clock goes from 3 to 6, it revolves by 900 or 1 right angle.
(b) If the hour hand of a clock goes from 2 to 8, it revolves by 1800 or 2 right angles.
(c) If the hour hand of a clock goes from 5 to 11, it revolves by 1800 or 2 right angles.
(d) If the hour hand of a clock goes from 10 to 1, it revolves by 900 or 1 right angle.
(e) If the hour hand of a clock goes from 12 to 9, it revolves by 2700 or 3 right angles.
(f) If the hour hand of a clock goes from 12 to 6, it revolves by 1800 or 2 right angles.
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?
We know that in 1 complete revolution in either a clockwise or anticlockwise direction, the hour hand of a clock will rotate by 3600 or 4 right angles.
(a) If the hour hand of a clock starts from 6 and turns through 1 right angle, it will stop at 9.
(b) If the hour hand of a clock starts from 8 and turns through 2 right angles, it will stop at 2.
(c) If the hour hand of a clock starts from 10 and turns through 3 right angles, it will stop at 7.
(d) If the hour hand of a clock starts from 7 and turns through 2 straight angles, it will stop at 7.
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?
We know that in one complete clockwise revolution, the hour hand will rotate by 3600
(a) When the hour hand of a clock starts at 12 and makes 1/2 of a revolution clockwise, it will rotate by 1800.
Hence, the hour hand of a clock will stop at 6.
(b) When the hour hand of a clock starts at 2 and makes 1/2 of a revolution clockwise, it will rotate by 1800
Hence, the hour hand of a clock will stop at 8.
(c) When the hour hand of a clock starts at 5 and makes 1/4 of a revolution clockwise, it will rotate by 900
Hence, the hour hand of a clock will stop at 8.
(d) When the hour hand of a clock starts at 5 and makes 3/4 of a revolution clockwise, it will rotate by 2700
Hence, the hour hand of a clock will stop at 2.
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?
By revolving one complete revolution either in a clockwise or anticlockwise direction, we will revolve by 3600, and two adjacent directions are at 900 or 1/4 of a complete revolution away from each other.
(a) If we start facing towards the east and turn clockwise to face north, we have to make 3/4 of a revolution.
(b) If we start facing towards the south and turn clockwise to face east, we have to make 3/4 of a revolution.
(c) If we start facing towards the west and turn clockwise to face east, we have to make 1/2 of a revolution.
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 know that in one complete clockwise revolution, 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
∴ Fraction = 1800/3600
= 1/2
(b) When the hour hand goes from 4 to 7 clockwise, it will rotate by 1 right angle or 900
∴ Fraction = 900/3600
= 1/4
(c) When the hour hand goes from 7 to 10 clockwise, it will rotate by 1 right angle or 900
∴ Fraction = 900/3600
= 1/4
(d) When the hour hand goes from 12 to 9 clockwise, it will rotate by 3 right angles or 2700
∴ Fraction = 2700/3600
= 3/4
(e) When the hour hand of a clock goes from 1 to 10 clockwise, it will rotate by 3 right angles or 2700
∴ Fraction = 2700/3600
= 3/4
(f) When the hour hand goes from 6 to 3 clockwise, it will rotate by 3 right angles or 2700
∴ Fraction = 2700/3600
= 3/4
How many right angles do you make if you start facing
(a) south and turn clockwise to the west?
(b) north and turn anticlockwise to the east?
(c) west and turn to the west?
(d) south and turn to the north?
By revolving one complete round in either a clockwise or anticlockwise direction, we will revolve by 3600, and two adjacent directions are 900 away from each other.
(a) If we start facing towards the south and turn clockwise to the west, we have to make one right angle.
(b) If we start facing towards the north and turn anticlockwise to east, we have to make 3 right angles.
(c) If we start facing towards the west and turn to the west, we have to make one complete round or 4 right angles.
(d) If we start facing towards the south and turn to the north, we have to make 2 right angles.
Match the following:
(i) Straight angle (a) Less than onefourth of a revolution
(ii) Right angle (b) More than half a revolution
(iii) Acute angle (c) Half of a revolution
(iv) Obtuse angle (d) Onefourth of a revolution
(v) Reflex angle (e) Between 1 / 4 and 1 / 2 of a revolution
(f) One complete revolution
(i) Straight angle = 1800 or half of a revolution
Hence, (c) is the correct answer
(ii) Right angle = 900 or onefourth of a revolution
Hence, (d) is the correct answer
(iii) Acute angle = less than 900 or less than onefourth of a revolution
Hence, (a) is the correct answer
(iv) Obtuse angle = more than 900 but less than 1800 or between 1 / 4 and 1 / 2 of a revolution
Hence, (e) is the correct answer
(v) Reflex angle = more than 1800 but less than 3600 or more than half a revolution
Hence, (b) is the correct answer
Classify each one of the following angles as right, straight, acute, obtuse or reflex:
(i) The given angle is an acute angle it measures less than 900
(ii) The given angle is an obtuse angle as it measures more than 900 but less than 1800
(iii) The given angle is a right angle as it measures 900
(iv) The given angle is a reflex angle as it measures more than 1800 but less than 3600
(v) The given angle is a straight angle as it measures 1800
(vi) The given angle is an acute angle as it measures less than 900
Find the measure of the angle shown in each figure. (First, estimate with your eyes and then find the actual measurement with a protractor).
The measures of the angles shown in the above figures are 400, 1300, 650 and 1350
Find the angle measure between the hands of the clock in each figure.
The angle measurement between the hands of the clock are 900, 300 and 1800
Investigate
In the given figure, the angle measure 300. Look at the same figure through a magnifying glass. Does the angle become larger? Does the size of the angle change?
The measure of an angle will not change by viewing through a magnifying glass.
From these two angles, which has a larger measure? Estimate and then confirm by measuring them.
The measures of these angles are 450 and 550. Hence, the angle shown in the second figure is greater.
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) 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 an obtuse angle (but less than 1800).
(c) An angle whose measure is the sum of the measures of two right angles is a straight angle.
(d) When the sum of the measures of two angles is that of a right angle, then each one of them is an acute angle.
(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 an obtuse angle.
What is the measure of
(i) a right angle?
(ii) a straight angle?
(i) The measure of a right angle is 900
(ii) The measure of a straight angle is 1800
Say True or False.
(a) The measure of an acute angle < 900
(b) The measure of an obtuse angle < 900
(c) The measure of a reflex angle > 1800
(d) The measure of one complete revolution = 3600
(e) If m ∠A = 530 and m ∠B = 350, then m ∠A > m ∠B.
(a) True, the measure of an acute angle is less than 900
(b) False, the measure of an obtuse angle is more than 900 but less than 1800
(c) True, the measure of a reflex angle is more than 1800
(d) True, the measure of one complete revolution is 3600
(e) True, ∠A is greater than ∠B
Write down the measures of
(a) some acute angles
(b) some obtuse angles
(Give at least two examples of each)
(a) The measures of an acute angle are 500, 650
(b) The measures of obtuse angle are 1100, 1750
Measures the angles given below using the protractor and write down the measure.
(a) The measure of an angle is 450
(b) The measure of an angle is 1200
(c) The measure of an angle is 900
(d) The measures of angles are 600, 900 and 1300
Which angle has a large measure? First, estimate and then measure.
The measure of Angle A =
The measure of Angle B =
The measure of angle A is 400
The measure of angle B is 680
∠B has a large measure than ∠A.
Measure and classify each angle.
Angle 
Measure 
Type 

∠AOB 

∠AOC 

∠BOC 

∠DOC 

∠DOA 

∠DOB 
Angle 
Measure 
Type 
∠AOB 
400 
Acute 
∠AOC 
1250 
Obtuse 
∠BOC 
850 
Acute 
∠DOC 
950 
Obtuse 
∠DOA 
1400 
Obtuse 
∠DOB 
1800 
Straight 
Which of the following are models for perpendicular lines?
(a) The adjacent edges of a tabletop
(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 tabletop 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 the letter V are inclined to form an acute angle.
Therefore, (a) and (c) are models for perpendicular lines.
Let be perpendicular to the line segment , and intersect in point A. What is the measure of ∠PAY?
From the figure, it is clear that the measure of ∠PAY is 900.
There are two set squares in a box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common?
The measure of angles in one set square are 300, 600 and 900.
The other set square has a measure of angles 450, 450 and 900.
Yes, the angle of measure 900 is common between them.
Study the diagram given below. 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, since CE = 2 units and EG = 2 units, CE = EG.
(b) Yes, since CE = EG, and both are of 2 units, PE bisect CG.
(c) and are the two line segments for which PE is the perpendicular bisector.
(d) (i) True, since AC = 2 units and FG = 1 unit.
(ii) True because both are of 1 unit.
(iii) True, since BC = 1 unit and EH = 3 units.
Name each of the following triangles in two different ways. (You may judge the nature of the angle by observation.)
(i) Acuteangled and isosceles triangle
(ii) Rightangled and scalene triangle
(iii) Obtuseangled and isosceles triangle
(iv) Rightangled and isosceles triangle
(v) Equilateral and acuteangled triangle
(vi) Obtuseangled and scalene triangle
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.
(c) ∆PQR, such that PQ = QR = PR = 5 cm.
(d) ∆DEF with ∠D = 90°
(e) ∆XYZ with ∠Y = 90° and XY = YZ.
(f) ∆LMN with ∠L = 30°, ∠M = 70° and ∠N = 80°.
(a) Scalene triangle
(b) Scalene triangle
(c) Equilateral triangle
(d) Rightangled triangle
(e) Rightangled isosceles triangle
(f) Acuteangled triangle
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.)
Name the type of triangle in each case. If you cannot make a triangle, think of reasons for it.
(a) By using three match sticks, we may make a triangle, as shown below.
The above triangle is an equilateral triangle.
(b) By using 4 match sticks, we cannot make a triangle since we know that sum of the lengths of any two sides of a triangle is always greater than the third side.
(c) By using 5 match sticks, we may make a triangle as shown below.
The above triangle is an isosceles triangle.
(d) By using 6 match sticks, we may make a triangle, as shown below.
The above triangle is an equilateral triangle.
Match the following.
Measures of Triangle  Type of Triangle 
(i) 3 sides of equal length  (a) Scalene 
(ii) 2 sides of equal length  (b) Isosceles rightangled 
(iii) All sides are of different lengths  (c) Obtuseangled 
(iv) 3 acute angles  (d) Rightangled 
(v) 1 right angle  (e) Equilateral 
(vi) 1 obtuse angle  (f) Acuteangled 
(vii) 1 right angle with two sides of equal length  (g) Isosceles 
(i) Equilateral triangle
(ii) Isosceles triangle
(iii) Scalene triangle
(iv) Acuteangled triangle
(v) Rightangled triangle
(vi) Obtuseangled triangle
(vii) Isosceles rightangled triangle
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 equal
(f) False, the opposite sides of a trapezium are not parallel
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?
A square is a regular quadrilateral because all the interior angles are 900, and all sides are of the same length.
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.
(a) A rectangle in which all the interior angles are of the same measure, i.e., 900 and only opposite sides of the rectangle are of the same length, whereas in a square, all the interior angles are 900, and all the sides of the square are of the same length. Hence, a rectangle with all sides equal becomes a square. Therefore, a square is a special rectangle.
(b) In a parallelogram, opposite sides are parallel and equal. In a rectangle, opposite sides are parallel and equal. The interior angles of the rectangle are of the same measure, i.e., 900. Hence, a parallelogram with each angle as a right angle becomes a square. Therefore, a rectangle is a special parallelogram.
(c) All sides of a rhombus and square are equal, but in the case of a square, all interior angles are 900. A rhombus in which each angle is a right angle becomes a square. Therefore, a square is a special rhombus.
(d) Since, all are closed figures with 4 line segments. Hence, all are quadrilaterals.
(e) Opposite sides of a parallelogram are equal and parallel, whereas in a square, opposite sides are parallel, and all 4 sides are of the same length. Therefore, a square is a special parallelogram.
Draw a rough sketch of a regular hexagon. Connecting any three of its vertices, draw a triangle. Identify the type of triangle you have drawn.
We can draw an isosceles triangle by joining three vertices of a hexagon, as shown in the below figure.
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.
The below figure is a regular octagon in which a rectangle is drawn by joining four of the vertices of the octagon.
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.
From the figure, we can find AC, AD, BD, BE and CE are the diagonals.
Examine whether the following are polygons. If any one among them is not, say why?
(i) It is not a closed figure. Hence, it is not a polygon.
(ii) It is a polygon made of six sides.
(iii) No, it is not a polygon because it is not made of line segments.
(iv) It is not a polygon, as it is not made of line segments.
Name each polygon.
Make two more examples of each of these.
(a) It is a closed figure and is made of four line segments. Hence, the given figure is a quadrilateral. Two more examples are
(b) The given figure is a triangle, as it is a closed figure with 3 line segments. Two more examples are
(c) The given figure is a pentagon, as the closed figure is made of 5 line segments. Two more examples are
(d) The given figure is an octagon, as it is a closed figure made of 8 line segments. Two more examples are
Match the following:
Give two new examples of each shape.
(a) An ice cream cone and a birthday cap are examples of the shape of a cone.
(b) A cricket ball and a tennis ball are examples of the shape of a sphere.
(c) A road roller and lawn roller are examples of the shape of a cylinder.
(d) A book and a brick are examples of the shape of a cuboid.
(e) A diamond and an Egypt pyramid are examples of the shape of a pyramid.
What shape is
(a) Your instrument box?
(b) A brick?
(c) A matchbox?
(d) A road roller?
(e) A sweet laddu?
(a) The shape of an instrument box is cuboid.
(b) The shape of a brick is cuboid.
(c) The shape of a matchbox is cuboid.
(d) The shape of a road roller is cylinder.
(e) The shape of a sweet laddu is sphere.
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