# R S AGGARWAL AND V AGGARWAL Solutions for Class 9 Maths Chapter 6 – Introduction to Euclid’s Geometry

## 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

(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

(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: (i) Three
lines

(ii) One
rectilinear figure

(iii) Four concurrent points

Solution 5 (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 is the same as line .

(v) A ray is the same as ray .

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

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

Solution 8  (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.

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) (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).  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 , if

(a) C is an interior point of AB

(b) AC = CB

(c) C is an interior point of AB such that = (d) AC + CB = AB

Solution 17

Correct option: (c)

A point C is called the midpoint of a line segment , if C is an interior point of AB such that = . 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) 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

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.

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