R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 6 - Introduction to Euclid's Geometry

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Chapter 6 - Introduction to Euclid's Geometry Exercise Ex. 6

Question 1

What is the difference between a theorem and an axiom?

Solution 1

A theorem is a statement that requires a proof. Whereas, a basic fact which is taken for granted, without proof, is called an axiom.

Example of Theorem: Pythagoras Theorem

Example of axiom: A unique line can be drawn through any two points.

Question 2

Define the following terms:

(i) Line segment (ii) Ray (iii) Intersecting lines (iv) Parallel lines (v) Half-line (vi) Concurrent lines (vii) Collinear points (viii) Plane

Solution 2

(i) Line segment: The straight path between two points is called a line segment.

(ii) Ray: A line segment when extended indefinitely in one direction is called a ray.

(iii) Intersecting Lines: Two lines meeting at a common point are called intersecting lines, i.e., they have a common point.

(iv) Parallel Lines: Two lines in a plane are said to be parallel, if they have no common point, i.e., they do not meet at all.

(v) Half-line: A ray without its initial point is called a half-line.

(vi) Concurrent lines: Three or more lines are said to be concurrent, if they intersect at the same point.

(vii) Collinear points: Three or more than three points are said to be collinear, if they lie on the same line.

(viii) Plane: A plane is a surface such that every point of the line joining any two points on it, lies on it.

Question 3

In the adjoining figure, name:

(i) Six points

(ii) Five line segments

(iii) Four rays

(iv) Four lines

(v) Four collinear points

Solution 3

(i) Six points: A,B,C,D,E,F

(ii) Five line segments:

(iii) Four rays:

(iv) Four lines:

(vi) Four collinear points: M,E,G,B

Question 4

In the adjoining figure, name:

(i) Two pairs of intersecting lines and their corresponding points of intersection

(ii) Three concurrent lines and their points of intersection

(iii) Three rays

(iv) Two line segments

Solution 4

(i) and their corresponding point of intersection is R.

and their corresponding point of intersection is P.

(ii) and their point of intersection is R.

(iii) Three rays are:

.

(iv) Two line segments are:

.

Question 5

From the given figure, name the following:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

(i) Three lines

(ii) One rectilinear figure

(iii) Four concurrent points

Solution 5

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

(i) Three lines: Line AB, Line PQ and Line RS

(ii) One rectilinear figure: EFGC

(iii) Four concurrent points: Points A, E, F and B

Question 6

(i) How many lines can be drawn to pass through a given point?

(ii) How many lines can be drawn to pass through two given points?

(iii) In how many points can the two lines at the most intersect?

(iv) If A, B, C are three collinear points, name all the line segments determined by them.

Solution 6

(i) An infinite number of lines can be drawn to pass through a given point.

(ii) One and only one line can pass through two given points.

(iii) Two given lines can at the most intersect at one and only one point.

(iv)

Question 7

Which of the following statements are true?

(i) A line segments has no definite length.

(ii) A ray has no end point.

(iii) A line has a definite length.

(iv) A line R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometryis the same as line R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry.

(v) A ray R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometryis the same as ray R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry.

(vi) Two distinct points always determine a unique line.

(vii) Three lines are concurrent if they have a common point.

(viii) Two distinct lines cannot have more than one point in common.

(ix) Two intersecting lines cannot be both parallel to the same line.

(x) Open half-line OA is the same thing as ray R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

(xi) Two lines may intersect in two points.

(xii) Two lines l and m are parallel only when they have no point in common.

Solution 7

(i) False

(ii) False

(iii) False

(iv) True

(v) False

(vi) True

(vii) True

(viii) True

(ix) True

(x) True

(xi) False

(xii) True

Question 8

In the given figure, L and M are mid-points of AB and BC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

(i) If AB = BC, prove that AL = MC.

(ii) If BL = BM, prove that AB = BC.

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

(ii) BL = BM

  2BL = 2BM

  AB = BC

Chapter 6 - Introduction to Euclid's Geometry Exercise MCQ

Question 1

In ancient India, the shapes of altars used for household rituals were

(a) squares and rectangles

(b) squares and circles

(c) triangles and rectangles

(d) trapeziums and pyramids

Solution 1

Correct option: (b)

Squares and circular altars were used for household rituals.

Whereas altars having shapes as combinations of rectangles, triangles and trapeziums were used for public worship.

Question 2

'Lines are parallel if they do not intersect' is started in the form of

(a) a definition

(b) an axiom

(c) a postulate

(d) a theorem

Solution 2

Correct option: (a)

'Lines are parallel if they do not intersect' is started in the form of a definition.

Question 3

Euclid stated that 'All right angles are equal to each other' in the form of

(a) a definition

(b) an axiom

(c) a postulate

(d) a proof

Solution 3

Correct option: (c)

Euclid stated that 'All right angles are equal to each other' in the form of a postulate.

This is Euclid's Postulate 4.

 

Note: The answer in the book is option (a). But if you have a look at the Euclid's postulate, the answer is a postulate.

 

Question 4

A pyramid is a solid figure, whose base is

(a) only a triangle

(b) only a square

(c) only a rectangle

(d) any polygon

Solution 4

Correct option: (d)

A pyramid is a solid figure, whose base is any polygon.

Question 5

The side faces of a pyramid are

(a) triangles

(b) squares

(c) trapeziums

(d) polygons

Solution 5

Correct option: (a)

The side faces of a pyramid are triangles.

Question 6

The number of dimensions of a solid are

(a) 1

(b) 2

(c) 3

(d) 5

Solution 6

Correct option: (c)

A solid has 3 dimensions.

Question 7

The number of dimensions of a surface is

(a) 1

(b) 2

(c) 3

(d) 0

Solution 7

Correct option: (b)

A surface has 2 dimensions.

Question 8

How many dimensions dose a point have

(a) 0 dimension

(b) 1 dimension

(c) 2 dimension

(d) 3 dimension

Solution 8

Correct option: (a)

A point is an exact location. A fine dot represents a point. So, a point has 0 dimensions.

Question 9

Boundaries of solids are

(a) lines

(b) curves

(c) surfaces

(d) none of these

Solution 9

Correct option: (c)

Boundaries of solids are surfaces.

Question 10

Boundaries of surfaces are

(a) lines

(b) curves

(c) polygons

(d) none of these

Solution 10

Correct option: (b)

Boundaries of surfaces are curves.

Question 11

The number of planes passing through three non-collinear points is

(a) 4

(b) 3

(c) 2

(d) 1

Solution 11

Correct option: (d)

The number of planes passing through three non-collinear points is 1.

Question 12

In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for

(a) household rituals

(b) public rituals

(c) both (a) and (b)

(d) none of (a), (b) and (c)

Solution 12

Correct option: (b)

In ancient India, altars with combination of shapes like rectangles, triangles and trapeziums were used for public rituals. 

Question 13

Axioms are assumed

(a) definitions

(b) theorems

(c) universal truths specific to geometry

(d) universal truths in all branches of mathematics

Solution 13

Correct option: (d)

Axioms are assumed as universal truths in all branches of mathematics because they are taken for granted, without proof.

Question 14

Which of the following is a true statement?

(a) The floor and a wall of a room are parallel planes

(b) The ceiling and a wall of a room are parallel planes.

(c) The floor and the ceiling of a room are the parallel planes.

(d) Two adjacent walls of a room are the parallel planes.

Solution 14

Correct option: (c)

Two lines are said to be parallel, if they have no point in common.

Options (a), (b) and (d) have a common point, hence they are not parallel.

In option (c), the floor and the ceiling of a room are parallel to each other is a true statement.

 

Question 15

Which of the following is true statement?

(a) Only a unique line can be drawn to pass through a given point

(b) Infinitely many lines can be drawn to pass through two given points

(c) If two circles are equal, then their radii are equal

(d)A line has a definite length.

Solution 15

Correct option: (c)

In option (a), infinite number of line can be drawn to pass through a given point. So, it is not a true statement.

In option (b), only one line can be drawn to pass through two given points. So, it is not a true statement.

In option (c),

'If two circles are equal, then their radii are equal' is the true statement.

In option (d), A line has no end points. A line has an indefinite length. So, it is not a true statement.

Question 16

Which of the following is a false statement?

(a) An infinite number of lines can be drawn to pass through a given point.

(b) A unique line can be drawn to pass through two given points.

(c) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

(d)A ray has one end point.

Solution 16

Correct option: (c)

Option (a) is true, since we can pass an infinite number of lines through a given point.

Option (b) is true, since a unique line can be drawn to pass through two given points.

Consider option (c).

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

 

As shown in the above diagram, a ray has only one end-point. So, option (d) is true.

 

Hence, the only false statement is option (c).

Question 17

A point C is called the midpoint of a line segment R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry, if

(a) C is an interior point of AB

(b) AC = CB

(c) C is an interior point of AB such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry=R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

(d) AC + CB = AB

Solution 17

Correct option: (c)

A point C is called the midpoint of a line segment R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry, if C is an interior point of AB such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry=R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry.

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry 

 

Question 18

A point C is said to lie between the points A and B if

(a) AC = CB

(b) AC + CB = AB

(c) points A, C and B are collinear

(d) options (b) and (c)

 

* Options modified

Solution 18

Correct option: (d)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 9 Mathematics Chapter - Introduction To Euclids Geometry

 

Observe the above figure. Clearly, C lies between A and B if AC + CB = AB.

That means, points A, B, C are collinear.

Question 19

Euclid's which axiom illustrates the statement that when x + y = 15, then x + y + z = 15 + z?

(a) first

(b) second

(c) third

(d) fourth

Solution 19

Correct option: (b)

Euclid's second axiom states that 'If equals are added to equals, the wholes are equal'.

Hence, when x + y = 15, then x + y + z = 15 + z. 

Question 20

A is of the same age as B and C is of the same age as B. Euclid's which axiom illustrates the relative ages of A and C?

(a) First axiom

(b) Second axiom

(c) Third axiom

(d) Fourth axiom

Solution 20

Correct option: (a)

Euclid's first axiom states that 'Things which are equal to the same thing are equal to one another'.

That is,

A's age = B's age and C's age = B' age

A's age = C's age 

Question 21

The number of interwoven isosceles triangles in Sriyantra is

(a) five

(b) seven

(c) nine

(d) eleven

Solution 21

Correct option: (c)

The Sriyantra consists of nine interwoven isosceles triangles.

Question 22

In Indus Valley Civilization (about 300 BC) the bricks used for construction work were having dimensions in the ratio

(a) 5:3:2

(b) 4:2:1

(c) 4:3:2

(d) 6:4:2

Solution 22

Correct option: (b)

In Indus Valley Civilization (about 300 BC) the bricks used for construction work were having dimensions in the ratio is 4:2:1.

 

Question 23

Into how many chapters was the famous treatise, 'The Elements' divided by Euclid?

(a) 13

(b) 12

(c) 11

(d) 9

Solution 23

Correct option: (a)

The famous treatise 'The Elements' was divided into 13 chapters by Euclid.

Question 24

Euclid belongs to the country

(a) India

(b) Greece

(c) Japan

(d) Egypt

Solution 24

Correct option: (b)

Euclid belongs to the country, Greece.

Question 25

Thales belongs to the country

(a) India

(b) Egypt

(c) Greece

(d) Babylonia

Solution 25

Correct option: (c)

Thales belongs to the country, Greece.

Question 26

Pythagoras was a student of

(a) Euclid

(b) Thales

(c) Archimedas

(d) Bhaskara

Solution 26

Correct option: (b)

Pythagoras was a student of Thales.

Question 27

Which of the following needs a proof?

(a) axiom

(b) postulate

(c) definition

(d) theorem

Solution 27

Correct option: (d)

A statement that requires a proof is called a theorem.