WBBSE Notes For Class 6 Maths Geometry Chapter 3 Geometrical Bos Its Instruments And Their Uses

Geometry Chapter 3 Geometrical Bos Its Instruments And Their Uses

Geometry Chapter 3 Names of the different instruments of a geometrical box

1. A geometrical box contains the following instruments:
   1. A ruler or scale
   2. A pair of dividers
   3. A pencil compass
   4. Two set squares
   5. A protractor.
2. In addition to these instruments, a geometrical box contains a pencil, an eraser for erasing wrong writings (written with pencils) or drawings, and a pencil cutter.

 

Geometry Chapter 3 Description of the instruments and their uses

A ruler of scale:

1. The standard ruler which is contained in the geometrical box is of length 6 inches or 15 centimetres.
2. One side of the ruler is marked in centimetres and millimetres and the other side is marked in inches.
3. The number of divisions in each centimetre or inch is 10.
4. With the help of a ruler or scale, we usually draw a line segment and also measure the length of a line segment.
5. It is also used to draw a line segment by joining two given points.
6. The diagram of a ruler is given below:
7. On the inch side, we find the mark 0 on the extreme left and 1, 2, 3, 4, 5, and 6 are marked on this side.
8. Each inch is divided into 10 equal parts.
9. On the other side i.e., on the centimetre side 0, 1, 2, 3, and 15 are marked and each centimetre is divided into 10 equal parts; each of these 10 sub-divided parts denotes 1 millimetre.

1. 

1. Suppose, you have to determine the length of any segment, AB, with the help of a ruler.
2. First of all place the scale upon the line AB such that the O mark (on the centimetre side) coincides with A while the other extremity B goes beyond 8 and point B falls on the 5 small marks after 8 cm.
3. Therefore the length of segment AB is 8-5 cm.

WBBSE Class 6 Regular Solids Notes

2.

1. Suppose, you have to draw a line segment of length equal to that of AB (= 6-5 cm). For this, first measure the length of the line segment AB, and let this length be 6-5 cm.
2. Then place the ruler on the plane of the paper where the line segment is to be Hold the ruler with the left hand.
3. Now taking the pencil in the right hand, put a point on the paper at the O mark with the sharp end of the pencil at the left and starting from there construct a line segment by drawing the pencil up to the marks 6 and five small markings after 6.
4. The length of the segment thus drawn would be equal to that of AB = 6-5 cm.

 

 

3. 

1. Suppose, a line segment of any length is to be drawn with the help of a ruler.
2. If we want to construct a line segment of length 3-8 cm (say), place the ruler on the plane of the paper where the line segment is to be drawn.
3. Then put the sharp end of the pencil at the O mark (on the cm-side) of the ruler and then from there move your pencil along the side of the ruler up to the mark 3 cm and 8 small markings after 3 cm.
4. The line segment thus drawn is of length 3-8 cm.

 

4.

1. Suppose, you have to join two given points on a plane of the paper so that a straight line segment is obtained.
2. Place the ruler such that it lies just below the given two points.
3. Then putting the sharp end of the pencil at one point which lies on the left side, move the pencil from there to the right side given point along the side of the ruler.
4. Thus we get a line segment joining the two given points.
5. In the same way, we can also extend the line on both sides of the given points.

Uses :

1. We use a ruler or scale to measure the length of a line segment.
2. A line segment of a given length can be drawn with the help of a ruler or scale.
3. It is also used to draw a line segment by joining two given points.
4. To measure the length, breadth and height of any regular body, a ruler or scale is used.
5. To draw different geometrical like angles, triangles, quadrilaterals etc., we use a ruler or scale.

Important Definitions Related to Geometrical Solids

A pair of dividers :

1. A pair of dividers is an instrument which looks like pincers with a pair of legs of equal length.
2. The lower end of each of the legs contains a needle and the upper ends of the legs are thicker and are fixed together with a  screw.
3. The lower ends of the legs i.e. the needles may be drawn apart according to our requirements.

 

 

Uses :

1. The distance between two points can be determined with the help of dividers.
2. A line of any given length can be drawn with the help of dividers.
3. With the help of dividers a line segment of length equal to that of a given line segment can be drawn.
4. We can cut a given required segment from a line segment of greater length with the help of dividers.

1.

1. Suppose, we have to determine the distance between two given points.
2. Place the needle points of the dividers upon the two given points.
3. Without disturbing the distance between the needle points by keeping the dividers fixed, place the dividers on a ruler such that one end of a needle be at the O-mark of the ruler and read the mark where the end of the other needle falls.
4. This gives the distance of the given points.

 

2.

1. Suppose, you have to draw a line segment of length 3 cm.
2. At first, place one of the needle points of the dividers on any of the markings on the ruler and the other needlepoint of the dividers is drawn apart in such a way that it reaches up to the 5th small marking beyond the 4 bold markings.
3. Then lift the dividers without disturbing the distance between the needles, and mark two points by pressing the dividers on the paper.
4. Join these two points with the ruler.
5. Thus you get a line segment of length 3 cm.

 

 

3. 

1. Suppose, AB is a given segment of a certain straight line of length equal to that of AB.
2. First, measure the distance between points A and B with the help of the dividers, and without disturbing the dividers, construct two points C and D’ by putting the needle points of the dividers on the plane of the paper where the required straight line is to be drawn.
3. Now join the CD with help of a pencil and a ruler.
4. Then CD gives the required straight line whose length is equal to that of AB.

 

 

4. 

1. Suppose, you have to cut a part whose length is equal to CD from a line segment AB.
2. In the first place, the needle points one at C and the other at D.
3. Now lift the dividers and without disturbing the distance between the needle points of the dividers put one needlepoint at A and let the other needlepoint of the dividers falls at E on the line segment AB.
4. So AE is equal to the length CD.

 

 

5.

1. We also use a pair of dividers in different ways to draw different geometrical such as angles, triangles, quadrilaterals and circles etc.
2. To cut a given, length of a line segment or to replace a definite length of the line segment.

Understanding Geometrical Solids

A pencil compass

1. A pencil compass has two legs.
2. The lower end of one leg contains a needle (like dividers) and on the other leg, a pencil can be fixed with a screw.
3. A pencil compass is used to draw circles.
4. In order to draw a circle, the distance between the needle point and the end of the pencil is adjusted such that this distance is equal to the radius of the circle.
5. Now place the needle point on the plane of the.
6. paper where the circle is to be drawn.
7. Then holding the pencil compass with the right hand at the top of the instrument and keeping the needlepoint fixed, move the end of the pencil on the plane of the paper around the fixed needlepoint.
8. The bounded so drawn is the circle.
9. Here the fixed point of the needle is the centre of the circle.

 

 

Uses :

1. To draw a circle, semicircle, or arc of a circle we use a pencil compass.
2. To draw different angles without using a protractor, a pencil compass is used.
3. To draw an angle equal to another angle a pencil compass is necessary.
4. We also use pencil compasses to draw different geometrical like triangles or, quadrilaterals having different lengths of the line segment.

 

Two set squares

1. There are two set squares in the geometrical box.
2. They are of different sizes in angles and also insides.
3. One set square has angles of 30°, 60° and 90°; the length of the largest side is twice that of the smallest side.
4. The other set square has angles 45°, 45° and 90°; two sides of it are of equal lengths.
5. The largest side in each of the set squares is called the hypotenuse.

Uses:

1. We can draw angles 30°, 45°, 60° and 90° with the help of set squares.
2. With the help of set squares, we can draw a line perpendicular to another line.
3. With the set squares, we can draw a line parallel to another line.

 

1.

1. Suppose, you have to draw angles 30°, 45°, 60° and 90°.
2. These angles can be drawn with the set squares.
3. Place one set square which has angles 30°, 60° and 90° on the plane of the paper.
4. Then by drawing a pencil through its border, we get a triangle.
5. Its angles are 30°, 60° and 90°.
6. If you require individual angles, then only by drawing the pencil through the border of two sides pairwise, do you get the required angles.
7. In the same way, with another set square you can draw angles 45° and 90°.

 

 

2.

1. Suppose, you have to draw a line perpendicular to a point on another line.
2. First, draw the straight line on which the perpendicular line is to be drawn.
3. Let AB be the straight line and O be a point at which the perpendicular line on AB is to be drawn.
4. Now place the set square so that one of its sides containing the right angle coincides with AB.
5. Then move the set square along the line AB towards the point O such that the other side containing the right angle i.e., the vertical side falls at O or in other words the other side containing the right angle lies at O vertically on AB.
6. Then a line OM is drawn along the border of the vertical side of the set square.
7. Lift the set square.
8. ∴ OM is drawn perpendicular to AB.

 

3.

1. Suppose, you have to draw a line through a given point parallel to a given line.
2. This construction can be done with the help of two set squares.
3. Let AB be the given line and C be the given point.
4. You have to draw a line through C parallel to AB.
5. Point C lies outside AB.
6. First of all, place a set square on line AB such that one of its sides containing the right angle coincides with AB.
7. Now hold this set square with your left hand and place another set square in such a way that one of its sides containing the right angle lies horizontally above AB and the other side containing the right angle towards the vertical side of the former set square and touches it as shown in the below.
8. Then move the second set square upwards till the horizontal side of it touches C.
9. Now a straight line is drawn along the border of the horizontal side of the second set square through C.
10. Let this line be CD. Lift both the set squares.
11. The line CD is drawn parallel to AB.
12. We also use two set squares to draw the angles 75°, 105°, 120°, 135°, 150°, and 180° other than the standard angles as stated in (1).
13. With the help of two set squares, we can easily draw geometrical like isosceles triangles, scalene triangles, isosceles right-angled triangles, scalene right-angled triangles, squares, rectangles, parallelograms, trapeze omes, rhombuses, etc.

 

 

 

 

 

 

 

 

 

 

 

Protractor:

1. Protractor is a very useful instrument.
2. It is a semicircular type; its circumference is divided into 180 equal parts.
3. There is a mark C at the centre of this instrument.
4. The protractor is marked from each end and the markings are given from 0° to 180° in both the clockwise and anticlockwise directions.

Uses:

1. A protractor is used
2. To draw an angle of a given measurement
3. To measure a given angle.

1.

1. Suppose, you have an angle equal to 65°.
2. First, you draw a straight line AB on the plane of the paper.
3. Mark a point C on AB.
4. Now place the protractor on AB such that the centre of it falls on C and the 0°-180° line coincides with AB; the semicircular portion lies above AB.
5. Now mark a point P on the paper against the mark 65° (60° and 5 small markings after it) on the protractor starting from 0° on AB towards the right-hand side.
6. Now remove the protractor and draw the straight line PC by joining the points P and C.
7. Then ∠PCB is the required angle i.e., ∠PCB = 65°.

 

 

2. 

1. Suppose, you have to measure an angle ∠POQ.
2. The protractor is placed on the angle ∠POQ such that the centre of the protector falls on O and the 0º – 180º line coincides with the line OQ.
3. You see that the arm OP falls along the mark of 60º on the circumference of the protractor.
4. So the angle POQ measures the angle 60º
5. ∴ ∠POQ = 60º

 

 

Geometry Chapter 3 Some Geometrical Figures

 

Angle:

1. When two line segments lying in the same plane intersect at a point, an angle is formed at their point of intersection.
2. The line segments are called arms.
3. Here, in ∠PQR is an angle.
4. PQ and QR are its arms and point Q is said to be the vertex of the angle.

 

 

Different types of angles :

Acute angle:

1. An angle which is less than a right angle or 90° is called an Acute Angle.
2. In ∠AOB is less than 90° i.c, a right angle and so ∠AOB is an acute angle.
3. 30°, 60°, 75° etc. arc acute angles.

 

 

obtuse Angle:

1. An angle which is greater than 90° but less than 180° is called an obtuse angle.
2. In ∠PQR is greater than 90° but less than 180°.
3. Therefore ∠PQR is an obtuse angle.
4. 100°, 120°, 135°, 175° etc. are obtuse angles.

 

 

Right Angle:

1. Two straight lines are such that one stands on the other at the point and the two adjacent angles formed are equal to one another, then each of these two adjacent angles is called a right angle.
2. 1 right angle =; 90° (90 degrees).
3. Here ∠AOB = 90°.

 

 

Reflex Angle:

1. An angle which is greater than two right angles i.e., 180º but less than four right angles i.e. 360° is called a Reflex angle.
2. In the above, the indicated angles are reflex angles.
3. 200°, 300°, 330°, 34.5° etc. are reflex angles.

Real-Life Scenarios Involving Architecture and Design

Straight Angle:

1. An angle which is exactly equal to two right angles i.e., an angle whose two arms lie in opposite directions in a straight line is called a straight angle.
2. ∠AOB = 180°
3. 1 straight angle = 180° = 2 x 90° = 2 right angles.

 

 

Triangle:

1. A triangle is a plane bounded by three
2. line segments which are obtained by joining three non-collinear points in the plane.
3. Here ΔABC is a triangle.
4. Its three arms are AB, BC and CA and its three angles are ∠ABC, ∠BAC, and ∠ACB.
5. It has three vertices A, B and C.
6. In any triangle, the sum of three angles of it is 180°.
7. ∠A + ∠B + ∠C = 180°.

 

 

On the basis of the sides, the triangles are divided into three classes:

1. Scalene triangle
2. Isosceles triangle and
3. Equilateral triangle.

 

Scalene triangle:

1. If all three sides of a triangle are of different lengths, then the triangle is called a scalene triangle.
2. In the AB ≠ BC ≠ CA.
3. So the triangle ABC is a scalene triangle.
4. It is also seen that ∠ABC ≠ ∠BAC ≠ ∠ACB.

 

 

Isosceles Triangle:

1. If the lengths of two sides of a triangle are equal, then the triangle is called an isosceles triangle.
2. In ΔABC is an isosceles triangle because the lengths of the sides AB and AC are equal.
3. AB = AC
4. In an isosceles triangle, the opposite angles of equal sides are equal.
5. The opposite angles of equal sides AB and AC are ∠ACB and ∠ABC respectively.
6. ∴ ∠ACB = ∠ABC.

 

 

 

Equilateral triangle:

1. If the lengths of all three sides of a triangle are equal then the triangle is called an equilateral triangle.
2. In an equilateral triangle, all the angles of the triangle are also equal and each is equal to 60°.
3. In ΔABC is an equilateral triangle because, AB = BC = CA i.e., all three sides are of equal length.
4. Again, ∠BAC = ∠ABC = ∠ACB = 60°.

 

 

Again on the basis of angles, triangles are divided into three classes:

1. Acute-angled triangle
2. Obtuse-angled triangle and
3. Right-angled triangle.

 

Acute-angled triangle:

1. If all three angles of a triangle are acute angles, then the triangle is called an acute-angled triangle.
2. In the ΔABC is an acute-angled triangle because all three angles ∠BAC, ∠ABC and ∠ACB are acute angles.

 

 

Obtuse angled triangle:

1. If one angle of a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle.
2. In ΔABC is an obtuse-angled triangle because ∠ABC is an obtuse angle.
3. Each of the angles ∠BAC and ∠ACB is an acute angle.

 

 

Right-angle triangle:

1. If one angle of a triangle is a right angle i.e., 90°, then the triangle is called a right-angled triangle.
2. The ΔABC is a right-angled triangle because ∠ABC = 90° i.e., a right angle and∠BAC, ∠ACB are acute angles.
3. Here AC is the opposite side of ∠ABC = 90°.
4. So AC is called the hypotenuse of the triangle ABC.
5. AB is perpendicular to the BC at B.
6. Then
7. This is called Pythagoras Theorem.

 

Quadrilateral:

1. A plane bounded by four line segments is called a quadrilateral.
2. Its 4 sides are AB, BC, CD and DA; 4 angles are ∠DAB, ∠ABC, ∠BCD and ∠CDA.
3. The sum of the four angles of a quadrilateral is 360°.
4. ∴∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
5. The vertices of the quadrilateral are A, B, C, and D.

 

 

 

 

Different types of Quadrilaterals :

Parallelogram:

1. A quadrilateral is said to be a Parallelogram if its opposite sides are parallel.
2. The PQRS is a parallelogram because PQ || SR and PS || QR.
3. In the parallelogram, the opposite sides are equal and also the opposite angles are equal.
4. Here PQ = SR and PS = QR ; ∠PSR = ∠PQR and ∠SPQ = ∠QRS.
5. The diagonals of the parallelogram are PR and SQ and PR ≠ SQ.
6. The diagonals of parallelograms bisect each other.
7. SO = OQ and PO = OR.

 

 

Rectangle:

1. A quadrilateral is said to be a Rectangle if the opposite sides are equal and each of the angles is one right.
2. The ABCD is a rectangle because AB = DC and AD = BC and ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°.
3. Its diagonals AC and BD are equal i.e., AC = BD and they bisect each other at O i.e., AO = OC = BO = DO.
4. Here AB || DC and AD || BC.
5. The opposite sides are parallel to one another.

 

 

Square:

1. A quadrilateral is said to be a square if all the sides of it are equal to one another and each of the angles of it is a right angle.
2. In the ABCD is a square because AB = BC = CD = DA and ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°.
3. The diagonals AC and BD are equal i.e., AC = BD and the diagonals bisect each other.
4. AO = OC – OB = OD.
5. Here AB || DC and BC i.e., the opposite sides are parallel to one another.

 

 

Rhombus:

1. A quadrilateral is said to be a Rhombus if all the sides of it are equal to one another and none of its angles is a right angle.
2. In ABCD is rhombus because AB = BC = CD = DA and none of its angles ∠ABC, ∠BCD, ∠CDA, or ∠DAB is a right angle.
3. Here the diagonals are not equal i.e. AC ≠ BD but AO = OC, BO = OD.
4. Here AB || DC and AD || BC.
5. The diagonals bisect each other at right angles.
6. ∴ ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°.

 

 

Trapezium:

1. If only one pair of opposite sides of a quadrilateral are parallel but not equal, then it is called a trapezium.
2. The remaining two opposite sides which are not parallel are called oblique sides.
3. In the ABCD is a trapezium.
4. It’s one pair of opposite sides AB and DC are parallel; AD and BC are oblique sides.

 

 

1. Isosceles trapezium:
2. If the length of two non-parallel sides i.e., the oblique sides of a trapezium are equal, then it is called an isosceles trapezium.
3. In the ABCD is an isosceles trapezium.
4. Its two opposite sides AB and DC are parallel and the lengths of the non-parallel sides i.e., the oblique sides AD and BC are equal i.e., AD = BC.