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.