Lesson: Introduction to Euclid's Geometry

Exercise 5.1 (22 Multiple Choice Questions and

Answers)

Question: 1

The three steps from solids to points are:

(a) Solids − surfaces − lines − points

(b) Solids − lines − surfaces − points

(c) Lines − points − surfaces − solids

(d) Lines − surfaces − points − solids

Solution:

a

Question 2

What is the dimension of a solid?

(a) 1

(b) 2

(c) 3

(d) 0

Solution:

c

Question 3

What is the dimension of a surface?

(a) 1

(b) 2

(c) 3

(d) 0

Solution:

b

Question 4

What is the dimension of a point?

(a) 0

(b) 1

(c) 2

(d) 3

Solution:

a

Question 5

Euclid divided his famous treatise “Elements” into:

(a) 13 chapters

(b) 12 chapters

(c) 11 chapters

(d) 9 chapters

Solution:

a

Question 6

How many propositions were deduced by Euclid?

(a) 465

(b) 460

(c) 13

(d) 55

Solution:

a

Question 7

Boundaries of solids are:

(a) surfaces.

(b) curves.

(c) lines.

(d) points.

Solution:

a

Question 8

Boundaries of surfaces are:

(a) surfaces.

(b) curves.

(c) solids.

(d) points.

Solution:

b

Question 9

In Indus Valley Civilisation (about 3000 B.C.), the

ratio length: breadth: thickness of the bricks, used for

constructions work, was:

(a) 1 : 3 : 4

(b) 4 : 2 : 1

(c) 4 : 4 : 1

(d) 4 : 3 : 2

Solution:

b

Question 10

A pyramid is a solid figure, the base of which is:

(a) Only a triangle

(b) Only a square

(c) Only a rectangle

(d) Any polygon

Solution:

d

A pyramid is a solid figure, the base of which is a

triangle or square or any other polygon.

Question 11

The side faces of a pyramid are:

(a) Triangles

(b) Squares

(c) Polygons

(d) Trapeziums

Solution:

a

The base of the pyramid may be any polygon but the

side faces of a pyramid are always triangles

.

Question 12

Which one of these statements illustrates that

“if x + y = 10 then x + y + z = 10 + z”?

(a) Things, which are equal to the same thing, are

equal to one another.

(b) If equals are added to equals, the wholes are equal.

(c) If equals are subtracted from equals, the

remainders are equal.

(d) The whole is greater than the part.

Solution:

b

The Euclid’s axiom that illustrates the given statement

is:

If equals are added to equals, the wholes are equal.

Question 13

In ancient India, the shapes of altars used for

household rituals were:

(a) Squares and circles

(b) Triangles and rectangles

(c) Trapeziums and pyramids

(d) Rectangles and squares

Solution:

a

In ancient India, squares and circular altars were used

for household rituals.

Question 14

The number of interwoven isosceles triangles in

Sriyantra (in the Atharvaveda) is:

(a) Seven

(b) Eight

(c) Nine

(d) Eleven

Solution:

c

The Sriyantra (in the Atharvaveda) consists of nine

interwoven isosceles triangles.

Question 15

Greek’s emphasized on:

(a) Inductive reasoning

(b) Deductive reasoning

(c) Both A and B

(d) Practical use of geometry

Solution:

b

Greek’s emphasized on deductive reasoning.

Question 16

In Ancient India, Altars with combination of shapes

like rectangles, triangles and trapeziums were used for:

(a) Public worship

(b) Household rituals

(c) Both A and B

(d) None of A and B

Solution:

a

In Ancient India, square and circular altars were used

for household rituals, while altars, whose shapes were

combinations of rectangles, triangles and trapeziums,

were required for public worship.

Question 17

Euclid belongs to:

(a) Babylonia

(b) Egypt

(c) Greece

(d) India

Solution:

c

Question 18

Thales belongs to:

(a) Babylonia

(b) Egypt

(c) Greece

(d) Rome

Solution:

c

Question 19

Pythagoras was a student of:

(a) Thales

(b) Euclid

(c) Both A and B

(d) Archimedes

Solution:

a

Question 20

Which of these needs a proof?

(a) Theorem

(b) Axiom

(c) Definition

(d) Postulate

Solution:

a

Question 21

Euclid stated that all right angles are equal to each

other in the form of:

(a) an axiom.

(b) a definition.

(c) a postulate.

(d) a proof.

Solution:

c

Euclid stated that all right angles are equal to each

other in the form of a postulate.

Question 22

‘Lines are parallel if they do not intersect’ is stated in

the form of:

(a) An axiom

(b) A definition

(c) A postulate

(d) A proof

Solution:

b

‘Lines are parallel, if they do not intersect’ is the

definition of parallel lines.

Exercise 5.2

Question 1

Write whether the following statements are True or

False? Justify your answer:

(i) Euclidean geometry is valid only for curved

surfaces.

(ii) The boundaries of the solids are curves.

(iii) The edges of a surface are curves.

(iv) The things which are double of the same thing are

equal to one another.

(v) If a quantity B is a part of another quantity A, then

A can be written as the sum of B and some third

quantity C.

(vi) The statements that are proved are called axioms.

(vii) “For every line l and for every point P not lying

on a given line l, there exists a unique line m passing

through P and parallel to l ” is known as Playfair’s

axiom.

(viii) Two distinct intersecting lines cannot be parallel

to the same line.

(ix) Attempts to prove Euclid’s fifth postulate using

the other postulates and axioms led to the discovery of

several other geometries.

Solution:

(i) False, because Euclidean geometry is valid for all

figures in a plane.

Examples: Line, points, intersecting lines etc.

(ii) False, because the boundaries of the solids are

surfaces.

Example:

The boundaries of a sphere are curved but the

boundaries of a cuboid are plane surfaces and the

boundaries of a cone are a plane and a curved surface.

(iii) False, because the edges of spherical surfaces are

curved but the edges of cuboidal surfaces are lines.

(iv) True, because it is one of Euclid’s axioms.

(v) True, because it is one of Euclid’s axioms.

Some of Euclid’s axioms:

1) Things which are equal to the same thing are equal

to one another.

2) If equals are added to equals, the wholes are equal.

3) If equals are subtracted from equals, the remainders

are equal.

4) Things which coincide with one another are equal

to one another.

5) The whole is greater than the part.

6) Things which are double of the same things are

equal to one another.

7) Things which are halves of the same things are

equal to one another.

Axiom (5) gives us the definition of ‘greater than’.

For example, if a quantity B is a part of another

quantity A, then A can be written as the sum of B and

some third quantity C. Symbolically,

A B

means that

there is some C such that

A = B + C

(vi) False, the statements that are proved are called

theorems.

Euclid used the term postulate for the assumptions that

were specific to geometry and otherwise called axioms.

A theorem is a mathematical statement whose truth

has been logically established. Common notions (often

called axioms), on the other hand, were assumptions

used throughout mathematics and not specifically

linked to geometry.

(vii) True, because it is an equivalent versions of

Euclid’s fifth postulate and it is known as Playfair’s

axioms.

(viii) True, because it is an equivalent versions of

Euclid’s fifth postulate.

Euclid’s five postulates:

Postulate 1: A straight line may be drawn from any

one point to any other point.

Postulate 2: A terminated line can be produced

indefinitely.

Postulate 3: A circle can be drawn with any centre and

any radius.

Postulate 4: All right angles are equal to one another.

Postulate 5: If a straight line, falling on two straight

lines, makes the interior angles on the same side of it

taken together less than two right angles, then the

two straight lines, if produced indefinitely, meet on

that side on which the sum of angles is less than two

right angles.

(ix) True

All attempts to prove the fifth postulate as a theorem

led to a great achievement in the creation of several

other geometries. These geometries are quite different

from Euclidean geometry and called non-Euclidean

geometry.

Exercise 5.3

Question 1

Two salesmen make equal sales during the month of

August. In September, each salesman doubles his sale

of the month of August. Compare their sales in

September.

Solution:

Let the equal sale of two salesmen in August be x. In

September each salesman doubles his sale of August.

Thus, sale of the first salesman is 2x and the sale of

second salesman is 2x. According to Euclid’s axioms,

things which are double of the same things are equal

to one another. So, in September their sales are again

equal.

Question 2

It is known that x + y = 10 and that x = z. Show that z

+ y = 10?

Solution:

We have, x + y = 10 and x = z …. (i)

According to Euclid’s axioms, if equals are added to

equals, the wholes are equal. ...(ii)

So, from statement (ii),

we get 2x + y = z 10... (iii)

From equation (i) and (iii), we get z + y = 10

Question 3

Look at the Fig. 5.3. Show that length of

AH

lengths of (AB BC CD).

Solution:

From the given figure, we have AB + BC + CD = AD

[AB, BC and CD are the parts of AD]

Here, AD is also a part of AH.

According to one of Euclid’s axioms, the whole is

greater than the part.

i.e.,

AH AD

So, the length

AH

the length of AB + BC + CD.

Question 4

In the Fig.5.4, we have AB = BC, BX = BY. Show

that AX = CY.

Solution:

We have,

AB = BC ….(i)

and BX =

BY .....(ii)

According to one of Euclid’s axioms, if equals are

subtracted from equals, the remainders are equal.

So, on subtracting equation (ii) from equation (i), we

get

AB − BX = BC – BY

AX CY

Question 5

In the Fig.5.5, we have X and Y are the mid-points of

AC and BC and AX = CY. Show that AC BC.

Solution:

Given that X is the mid–point of AC

1

AX =CX = AC

2

…………………. (i)

2AX =2CX =AC

and Y is the mid – point of BC

1

BY=CY= BC

2

………………… (ii)

2BY=2CY=BC

Also, given that AX

CY ………………… (iii)

According to Euclid’s axiom, things which are double

of the same things are equal to one another.

From (iii), we get

2AX = 2CY

AC BC

[From (i) and (ii)]

Question 6

In the Fig.5.6, we have

B

1

X = AB

2

,

B

1

Y= BC

2

and AB = BC. Show that BX = BY.

Solution:

Given that:

B

1

X = AB

2

2BX AB

… (i)

B

1

Y= BC

2

2BY BC

… (ii)

and AB = BC … (iii)

On putting the values from equations (i) and (ii) in

equation (iii), we get

2BX = 2BY

According to Euclid’s axioms, things which are

double of the same things are equal to one another.

∴ BX = BY

Question 7

In the Fig.5.7, we have

1 2, 2 3

. Show

that

1 3

.

Solution:

Given that:

1 2

… (i)

and

2 3

… (ii)

According to Euclid’s axioms, things which are equal

to the same thing are equal to one another.

From equations (i) and (ii), we get

1 3

Question 8

In the Fig. 5.8, we have

1 3

and

2 4

. Show

that

A C

.

Solution:

Given that:

1 3

… (i)

and

2 4

… (ii)

According to Euclid’s axioms, if equals are added to

equals, then wholes are also equal. On adding

equations (i) and (ii), we get,

1 2 3 4

A C

Question 9

In the Fig. 5.9, we have

ABC ACB, 3 4

.

Show that

1 2

.

Solution:

Given that:

ABC ACB

… (i)

and

4 3

… (ii)

According to Euclid’s axiom, if equals are subtracted

from equals, the remainders are equal.

On subtracting equation (ii) from equation (i), we get

ABC 4 ACB 3

1 2

Question 10

In the Fig. 5.10, we have AC = DC, CB = CE. Show

that AB = DE.

Solution:

Given that:

AC = DC ….(i)

and CB = CE ...(ii)

According to Euclid’s axiom, if equals are added to

equals, then wholes are also equal.

So, on adding equation (i) and equation (ii), we get

AC + CB = DC + CE

AB DE

Question 11

In the Fig. 5.11, if

1

OX XY

2

,

1

PX XZ

2

and

OX = PX, show that XY = XZ.

Solution:

Given that:

1

OX XY

2

2OX XY

… (i)

and

1

PX XZ

2

2PX XZ

… (ii)

and OX = PX … (iii)

According to Euclid’s axioms, things which are

double of the same things are equal to one another.

On multiplying equation (iii) by 2, we get

2OX = 2PX

XY XZ

[From (i) and (ii)]

Question 12

In the Fig.5.12:

(i) AB = BC, M is the mid-point of AB and N is the

mid point of BC. Show that AM = NC.

(ii) BM = BN, M is the mid-point of AB and N is the

mid

point of BC. Show that AB = BC.

Solution:

(i) Given that:

AB = BC … (i)

M is the midpoint of AB.

1

AM MB AB

2

… (ii)

and N is the midpoint of BC

1

BN NC BC

2

… (iii)

According to Euclid’s axioms, things which are halves

of the same things are equal to one another.

From Equation (i), we get AB = BC

On multiplying both the sides by

1

2

, we get

1 1

AB BC

2 2

AM NC

[Using (ii) and (iii)]

(ii) Given that:

BM = BN … (i)

M is the midpoint of AB.

1

AM BM AB

2

2AM 2BM AB

… (ii)

and N is the midpoint of BC

1

BN NC BC

2

2BN 2NC BC

… (iii)

According to Euclid’s axioms, things which are

doubles of the same things are equal to one another.

On multiplying both sides of the equation (i) by 2, we

get

2BM = 2BN

AB BC

[using (ii) and (iii)]

Exercise 5.4

Question 1

Read the following statement:

An equilateral triangle is a polygon made up of three

line segments out of which two line segments are

equal to the third one and all its angles are 60° each.

Define the terms used in this definition which you feel

necessary.

Are there any undefined terms in this? Can you justify

that all sides and all angles are equal in a

equilateral triangle.

Solution:

Terms:

A polygon is a closed figure bounded by three or

more line segments.

A line segment is part of a line with two end

points.

An angle in a figure is formed by two rays with

one common initial point.

An acute angle is an angle whose measure is

between

0

to

90

.

Here, the undefined terms are ‘line’ and ‘point’.

Justification of the statement:

All the angles of an equilateral triangle are 60° each

Justification:

Given: Two line segments are equal to the third-one.

Therefore, all three sides of an equilateral triangle are

equal (According to Euclid’s axiom, things which are

equal to the same thing are equal to one another).

Question 2

Study the following statement:

“Two intersecting lines cannot be perpendicular to the

same line”.

Check whether it is an equivalent version to the

Euclid’s fifth postulate.

[Hint: Identify the two intersecting lines l and m and

the line n in the above

statement.]

Solution:

Two equivalent versions of Euclid’s fifth postulate are:

For every line L and for every point P not lying

on L, there exists a unique line M passing

through P and parallel to L.

Two distinct intersecting lines cannot be parallel

to the same line.

From above two statements it is clear that the given

statement is not an equivalent version to the Euclid’s

fifth postulate.

Question 3

Read the following statements which are taken as

axioms:

(i) If a transversal intersects two parallel lines, then

corresponding angles are not necessarily equal.

(ii) If a transversal intersect two parallel lines, then

alternate interior angles are equal.

Is this system of axioms consistent? Justify your

answer.

Solution:

A system of axiom is called consistent, if there is no

statement which can be deduced from these axioms

such that it contradicts any axiom.

The statement, if a transversal intersects two parallel

lines, then each pair of corresponding angles are equal,

is a theorem. So, statement (i) is false and is not an

axiom.

Also, if a transversal intersects two parallel lines, then

each pair of alternate interior angles are equal. It is

also a theorem. So, statement (ii) is true

and is an axiom.

Thus, the first statement is false and the second one is

an axiom.

Hence, the given system of axioms is not consistent.

Question 4

Read the following two statements which are taken as

axioms:

(i) If two lines intersect each other, then the

vertically opposite angles are not equal.

(ii) If a ray stands on a line, then the sum of two

adjacent angles so formed is equal to

180

.

Is this system of axioms consistent? Justify your

answer.

Solution:

A system of axiom is called consistent, if there is no

statement which can be deduced from these axioms

such that it contradicts any axiom.

If two lines intersect each other, then the vertically

opposite angles are equal. It is a theorem, so,

statement (i) is false and is not an axiom.

If a ray stands on line, then the sum of two adjacent

angles so formed is equal to. It is an axiom. So,

statement (ii) is true and is an axiom.

Thus, the first is false and the second one is true.

Hence, the given system of axioms is not consistent.

Question 5

Read the following axioms:

(i) Things which are equal to the same thing are equal

to one another.

(ii) If equals are added to equals, the wholes are equal.

(iii) Things which are double of the same thing are

equal to one another.

Check whether the given system of axioms is

consistent or inconsistent.

Solution:

Some of the Euclid’s axioms are:

The things which are equal to the same thing are

equal to one another.

If equals be added to the equals, the wholes are

equal.

If equals be subtracted from equals, the

remainders are equals.

Things which coincide with one another are equal

to one another.

The whole is greater than the part.

Things which are double of the same thing are

equal to one another.

Things which are halves of the same thing are

equal to one another.

Given axioms are:

(i) Things which are equal to the same thing are equal

to one another.

(ii) If equals are added to equals, the wholes are equal.

(iii) Things which are double of the same things are

equal to one another.

Thus, the three axioms are Euclid’s axioms which do

not contradict any axioms. So, the given system of

axioms is consistent.