## NCERT Class VI Mathematics Chapter 5 Understanding Elementary Shapes

National Council of Educational Research and Training (NCERT) Book for Class VI

Subject: Mathematics

Chapter: Chapter 5 – Understanding Elementary Shapes

Class VI NCERT Mathematics Text Book Chapter 5 Understanding Elementary Shapes is given below.

**5.1 Introduction**

All the shapes we see around us are formed using curves or lines. We can see corners, edges, planes, open curves and closed curves in our surroundings. We organise them into line segments, angles, triangles, polygons and circles. We find that they have different sizes and measures. Let us now try to develop tools to compare their sizes.

**5.2 Measuring Line Segments**

We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments.

A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments.

To compare any two line segments, we find a relation between their lengths. This can be done in several ways.

**Think, discuss and write**

1. What other errors and difficulties might we face?

2. What kind of errors can occur if viewing the mark on the ruler is not proper? How can one avoid it?

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?

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.

**5.3 Angles – ‘Right’ and ‘Straight’**

You have heard of directions in Geography. We know that China is to the north of India, Sri Lanka is to the south. We also know that Sun rises in the east and sets in the west. There are four main directions. They are North (N), South (S), East (E) and West (W).

Do you know which direction is opposite to north?

Which direction is opposite to west?

Just recollect what you know already. We now use this knowledge to learn a few properties about angles.

Stand facing north.

Note that there is no special name for three-fourth of a revolution.

**EXERCISE 5.2**

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

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?

(d) south and make one full revolution?

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

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?

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

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?

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?

Suppose any shape with corners is given. You can use your RA tester to

test the angle at the corners.

Do the edges match with the angles of a paper? If yes, it indicates a

right angle.

Do you see that each one of them is less than one-fourth of a revolution?

Examine them with your RA tester.

If an angle is larger than a right angle, but less than a straight angle, it is called an** obtuse angle**. These are obtuse angles.

**EXERCISE 5.3**

1. Match the following :

(i) Straight angle (a) Less than one-fourth of 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 of a revolution

(v) Reflex angle (e) Between 1/4 and 1/2 of a revolution

(f) One complete revolution

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

**5.5 Measuring Angles**

The improvised ‘Right-angle tester’ we made is helpful to compare angles with a right angle. We were able to classify the angles as acute, obtuse or reflex.

But this does not give a precise comparison. It cannot find which one among the two obtuse angles is greater. So in order to be more precise in comparison, we need to ‘measure’ the angles. We can do it with a ‘protractor’.

**The measure of angle**

We call our measure, ‘degree measure’. One complete revolution is divided into 360 equal parts. Each part is a **degree**. We write 360° to say ‘three hundred sixty degrees’.

**Think, discuss and write**

How many degrees are there in half a revolution? In one right angle? In one straight angle?

How many right angles make 180°? 360°?

1. Place the protractor so that the mid point (M in the figure) of its

straight edge lies on the vertex B of the angle.

**EXERCISE 5.4**

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

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 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).

4. Measure the angles given below using the Protractor and write down the measure.

(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 _______.

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).

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

**5.6 Perpendicular Lines**

When two lines intersect and the angle between them is a right angle, then the lines are said to be **perpendicular**. If a line AB is perpendicular to CD, we

write AB ⊥ CD.

**Think, discuss and write**

If AB ⊥ CD, then should we say that CD ⊥ ABalso?

**Perpendiculars around us!**

You can give plenty of examples from things around you for perpendicular lines (or line segments). The English alphabet T is one. Is there any other alphabet which illustrates perpendicularity?

**EXERCISE 5.5**

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.

(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.

**5.7 Classification of Triangles**

Do you remember a polygon with the least number of sides? That is a triangle.

Let us see the different types of triangle we can get.

Observe the angles and the triangles as well as the measures of the sides carefully. Is there anything special about them?

**What do you find?**

- Triangles in which all the angles are equal.

If all the angles in a triangle are equal, then its sides are also …………..

- Triangles in which all the three sides are equal.

If all the sides in a triangle are equal, then its angles are…………. .

- Triangle which have two equal angles and two equal sides.

If two sides of a triangle are equal, it has ………….. equal angles. and if two angles of a triangle are equal, it has ……………. equal sides.

- Triangles in which no two sides are equal.

If none of the angles of a triangle are equal then none of the sides are equal.

If the three sides of a triangle are unequal then, the three angles are also…………. .

**Naming triangles based on sides**

A triangle having all three unequal sides is called a **Scalene Triangle** [(c), (e)].

A triangle having two equal sides is called an **Isosceles Triangle** [(b), (f)].

A triangle having three equal sides is called an **Equilateral Triangle** [(a), (d)].

Classify all the triangles whose sides you measured earlier, using these definitions.

**Naming triangles based on angles**

If each angle is less than 90°, then the triangle is called an acute angled triangle. If any one angle is a right angle then the triangle is called a right angled triangle.

If any one angle is greater than 90°, then the triangle is called an obtuse angled triangle.

(c) a right angled isosceles triangle.

(d) a scalene right angled triangle.

Do you think it is possible to sketch

(a) an obtuse angled equilateral triangle ?

(b) a right angled equilateral triangle ?

(c) a triangle with two right angles?

Think, discuss and write your conclusions.

**EXERCISE 5.6**

1. Name the types of following triangles :

(a) Triangle with lengths of sides 7 cm, 8 cm and 9 cm.

(b) ΔABC with AB = 8.7 cm, AC = 7 cm and BC = 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°.

**2. 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 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 |

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

2. Using four unequal sticks, as you did in the above activity, see if you can form a quadrilateral such that

(a) all the four angles are acute.

(b) one of the angles is obtuse.

(c) one of the angles is right angled.

(d) two of the angles are obtuse.

(e) two of the angles are right angled.

(f) the diagonals are perpendicular to one another.

**EXERCISE 5.7**

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.

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.

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?

**5.9 Polygons**

So far you studied polygons of 3 or 4 sides (known as triangles and quardrilaterals respectively). We now try to extend the idea of polygon to figures with more number of sides. We may classify polygons according to the number of their sides.

You can find many of these shapes in everyday life. Windows, doors, walls, almirahs, blackboards, notebooks are all usually rectanglular in shape. Floor tiles are rectangles. The sturdy nature of a triangle makes it the most useful shape in engineering constructions.

Look around and see where you can find all these shapes.

**EXERCISE 5.8**

1. Examine whether the following are polygons. If any one among them is not, say why?

Make two more examples of each of these.

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.

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.

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.

**5.10 Three Dimensional Shapes**

Here are a few shapes you see in your day-to-day life. Each shape is a solid. It is not a‘flat’ shape.

Name any five things which resemble a sphere.

Name any five things which resemble a cone.

**Faces, edges and vertices**

In case of many three dimensional shapes we can distinctly identify their faces, edges and vertices. What do we mean by these terms: Face, Edge and Vertex? (Note ‘Vertices’ is the plural form of ‘vertex’).

Consider a cube, for example.

Each side of the cube is a flat surface called a flat **face** (or simply a **face**). Two faces meet at a line segment called an **edge**. Three edges meet at a point called a **vertex**.

The cylinder, the cone and the sphere have no straight edges. What is the base of a cone? Is it a circle? The cylinder has two bases. What shapes are they? Of course, a sphere has no flat faces!

**Think about it.**

**EXERCISE 5.9**

1. Match the following :

Give two new examples of each shape.

2. What shape is

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

(c) A match box? (d) A road-roller?

(e) A sweet laddu?

**What have we discussed?**

1. The distance between the end points of a line segment is its length.

2. A graduated ruler and the divider are useful to compare lengths of line segments.

3. When a hand of a clock moves from one position to another position we have an example for an angle.

One full turn of the hand is 1 revolution.

A right angle is ¼ revolution and a straight angle is ½ a revolution . We use a protractor to measure the size of an angle in degrees.

The measure of a right angle is 90° and hence that of a straight angle is 180°. An angle is acute if its measure is smaller than that of a right angle and is obtuse if its measure is greater than that of a right angle and less than a straight angle. A reflex angle is larger than a straight angle.

4. Two intersecting lines are perpendicular if the angle between them is 90°.

5. The perpendicular bisector of a line segment is a perpendicular to the line segment that divides it into two equal parts.

6. Triangles can be classified as follows based on their angles:

10. We see around us many three dimensional shapes. Cubes, cuboids, spheres, cylinders, cones, prisms and pyramids are some of them.

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