Euclid's Definitions, Axioms and Postulates Questions in English

(N/A) In the figure given above,the line segment $AC$ coincides with the sum of the line segments $AB$ and $BC$.
According to Euclid's Axiom $(4)$,things which coincide with one another are equal to one another.
Since the segment $AB + BC$ coincides with the segment $AC$,it follows that:
$AB + BC = AC$
Note that in this proof,it is assumed that there is a unique line passing through two distinct points.

(N/A) In the statement above,a line segment of any length is given,say $AB$ [see Fig. $(i)$].
Here,you need to do some construction. Using Euclid's Postulate $3$,you can draw a circle with point $A$ as the centre and $AB$ as the radius [see Fig. $(ii)$]. Similarly,draw another circle with point $B$ as the centre and $BA$ as the radius. The two circles meet at a point,say $C$. Now,draw the line segments $AC$ and $BC$ to form $\Delta ABC$ [see Fig. $(iii)$].
So,you have to prove that this triangle is equilateral,i.e.,$AB = AC = BC$.
Now,$AB = AC$,since they are the radii of the same circle $(1)$.
Similarly,$AB = BC$ (Radii of the same circle) $(2)$.
From these two facts,and Euclid's axiom that things which are equal to the same thing are equal to one another,you can conclude that $AB = BC = AC$.
So,$\Delta ABC$ is an equilateral triangle.
Note that here Euclid has assumed,without mentioning anywhere,that the two circles drawn with centres $A$ and $B$ will meet each other at a point.

(A-D) $(i)$ False. If we mark a point $O$ on the surface of a paper,we can draw an infinite number of straight lines passing through $O$.
$(ii)$ False. There is one and only one line which passes through two distinct points $P$ and $Q$.
$(iii)$ True. According to Euclid's Postulate $2$,a terminated line can be produced indefinitely.
$(iv)$ True. If two circles are equal,their centers and boundaries coincide when superimposed,so their radii must be equal.
$(v)$ True. According to Euclid's axiom,things which are equal to the same thing are equal to one another.

(N/A) Parallel lines: Two lines in a plane are said to be parallel if they do not intersect each other,no matter how far they are extended in either direction.
Yes,there are other terms that need to be defined first:
$1$. Point: $A$ point is a location in space that has no dimension (no length,breadth,or thickness).
$2$. Line: $A$ line is a breadthless length that extends infinitely in both directions.
$3$. Plane: $A$ surface which has length and breadth only.
$4$. Intersection: Two lines are said to intersect if they have a common point.
$5$. Distance between lines: The perpendicular distance between two parallel lines is constant at all points.

(N/A) Yes,these postulates contain undefined terms such as: 'Point' and 'Line'.
These postulates are consistent because they deal with two different situations:
$(i)$ states that given two points $A$ and $B$,there is a point $C$ lying on the line segment between them.
$(ii)$ states that,given $A$ and $B$,you can take a point $C$ that does not lie on the line passing through $A$ and $B$.
No,these postulates do not follow from Euclid's postulates. However,they follow from the axiom: 'Given two distinct points,there is a unique line that passes through them.'

(N/A) Given: Point $C$ lies between $A$ and $B$ such that $AC = BC$.
To prove: $AC = \frac{1}{2} AB$.
Proof:
Since $C$ lies between $A$ and $B$,we have $AC + BC = AB$.
Given that $AC = BC$.
Substituting $AC$ for $BC$ in the equation $AC + BC = AB$,we get:
$AC + AC = AB$
$2 AC = AB$
Dividing both sides by $2$,we get:
$AC = \frac{1}{2} AB$.
Hence proved.

(N/A) Given that $C$ is a mid-point of line segment $AB$,so $AC = BC$.
Since $AC + BC = AB$,we can write $AC + AC = AB$,which implies $2AC = AB$,or $AC = \frac{1}{2} AB$.
To prove that every line segment has one and only one mid-point,let us assume that there are two distinct mid-points $C$ and $D$ for the line segment $AB$.
Since $C$ is a mid-point,$AC = \frac{1}{2} AB$ ............. $(1)$
Since $D$ is a mid-point,$AD = \frac{1}{2} AB$ ............. $(2)$
From $(1)$ and $(2)$,we have $AC = AD$.
This implies that points $C$ and $D$ must coincide.
Thus,every line segment has one and only one mid-point.

(N/A) We have,
$AC = BD$ [Given] .......... $(1)$
Since the point $B$ lies between $A$ and $C$,
$\therefore AC = AB + BC$ .......... $(2)$
Similarly,since the point $C$ lies between $B$ and $D$,
$\therefore BD = BC + CD$ .......... $(3)$
From $(1)$,$(2)$,and $(3)$,
$AB + BC = BC + CD$
Subtracting $BC$ from both sides (Euclid's axiom: If equals are subtracted from equals,the remainders are equal),
$\Rightarrow AB = CD$.

(B) In the given list of Euclid's axioms,Axiom $5$ states: 'The whole is greater than the part'.
This statement is true for all things and in all parts of the universe,regardless of the nature of the objects.
Since it holds true universally without any exceptions,it is considered a 'universal truth'.

(A) Take any line $l$ and a point $P$ not on $l$.
According to Playfair's axiom,which is equivalent to Euclid's fifth postulate,there exists a unique line $m$ passing through $P$ that is parallel to $l$.
By definition,the distance of a point from a line is the length of the perpendicular segment drawn from the point to the line.
Since the lines are parallel,the perpendicular distance from any point on line $m$ to line $l$ remains constant.
Therefore,these two lines are everywhere equidistant from one another,which is a direct consequence of the fifth postulate.

(N/A) Euclid's fifth postulate is often considered complex. An equivalent and easier-to-understand version is known as Playfair's Axiom,which states: '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 $l$.' Alternatively,it can be stated as: 'Two distinct intersecting lines cannot be parallel to the same line.'

(N/A) Yes,Euclid's fifth postulate implies the existence of parallel lines.
If a straight line $l$ falls on two lines $m$ and $n$ such that the sum of the interior angles on one side of $l$ is equal to two right angles $(180^{\circ})$,then by Euclid's fifth postulate,the lines $m$ and $n$ will not meet on this side of $l$.
Since the sum of the interior angles on the other side of the line $l$ will also be equal to two right angles $(180^{\circ})$,the lines will not meet on the other side either.
$\therefore$ The lines $m$ and $n$ never meet,which means they are parallel to each other.

(N/A) Perpendicular lines: Two lines $p$ and $q$ lying in the same plane are said to be perpendicular if they intersect at a right angle $(90^{\circ})$,and we write them as $p \perp q$.
Yes,there are other terms that need to be defined first:
$1$. Line: $A$ line is a breadthless length that extends indefinitely in both directions.
$2$. Plane: $A$ surface which has only length and breadth.
$3$. Angle: The inclination between two intersecting lines.
$4$. Right angle: An angle whose measure is $90^{\circ}$.

(N/A) 'line segment' is defined as a part of a line that has two distinct endpoints and contains all the points on the line between them.
Yes,there are other terms that need to be defined first,specifically 'point' and 'line'.
$1$. Point: $A$ point is an exact location in space that has no dimensions (no length,breadth,or height).
$2$. Line: $A$ line is a straight path that extends infinitely in both directions and has no thickness.

(N/A) Definition of the radius of a circle:
The distance from the center to a point on the circle is called the radius of the circle. In the figure,$P$ is the center and $Q$ is a point on the circle,then the line segment $PQ$ is the radius.
Other terms that need to be defined first:
$1$. Circle: $A$ circle is the set of all points in a plane that are at a fixed distance from a fixed point in the plane.
$2$. Center: The fixed point from which all points on the circle are at a fixed distance is called the center of the circle.
$3$. Point: $A$ point is that which has no part.
$4$. Plane: $A$ plane is a surface which lies evenly with the straight lines on itself.

(N/A) square is a quadrilateral in which all four angles are right angles and all four sides are equal.
To define a square,we need to define the following terms first:
$1$. Quadrilateral: $A$ simple closed figure made up of four line segments.
$2$. Line segment: $A$ part of a line with two end points.
$3$. Angle: The inclination between two rays having a common initial point.
$4$. Right angle: An angle whose measure is $90^{\circ}$.
In the figure,$PQRS$ is a square.

(A) Euclid's axioms are fundamental assumptions used in geometry. The list of Euclid's axioms is as follows:
$1$. Things which are equal to the same thing are equal to one another.
$2$. If equals be added to equals,the wholes are equal.
$3$. If equals be subtracted from equals,the remainders are equals.
$4$. Things which coincide with one another are equal to one another.
$5$. The whole is greater than the part.
Therefore,Euclid's second axiom is: 'If equals be added to equals,the wholes are equal.'

(B) Euclid's fifth postulate states that 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 $(180^{\circ})$,then the two straight lines,if produced indefinitely,meet on that side on which the sum of angles is less than two right angles. This is a fundamental axiom in Euclidean geometry that relates to the parallel postulate.

(C) According to Euclid's Axioms,specifically the axiom which states that 'Things which are double of the same things are equal to one another'.
If $x = 2a$ and $y = 2a$,then $x = y$.
Therefore,the things which are double of the same thing are equal.

(D) In mathematics,axioms (or postulates) are fundamental statements or propositions that are assumed to be true without proof.
Specifically,Euclid's axioms are considered to be universal truths that apply to all branches of mathematics,not just geometry.
Unlike theorems,which require logical proof based on axioms,axioms serve as the starting point for all mathematical reasoning.

(A) Let the age of John be $J$,the age of Mohan be $M$,and the age of Ram be $R$.
Given that $J = M$ and $R = M$.
According to Euclid's first axiom,'Things which are equal to the same thing are equal to one another'.
Since both $J$ and $R$ are equal to $M$,it follows that $J = R$.
Therefore,John and Ram are of the same age,which is illustrated by the First Axiom.

(A) According to Euclid's fifth postulate,if a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than $180^{\circ}$,then the two straight lines,if produced indefinitely,meet on that side on which the sum of angles is less than $180^{\circ}$.
In the given problem,the sum of the interior angles on one side is $120^{\circ}$.
Since $120^{\circ} < 180^{\circ}$,the lines will meet on the side where the sum of the interior angles is less than $180^{\circ}$.
However,the question asks for the condition relative to the given sum. Since the lines meet on the side where the sum is less than $180^{\circ}$,and we are given a specific side with a sum of $120^{\circ}$,the lines will meet on this side because $120^{\circ} < 180^{\circ}$.
Among the given options,the lines meet on the side where the sum is less than $180^{\circ}$. Since the question asks for the side where the sum is less than $180^{\circ}$,and $120^{\circ}$ is the given sum,the lines meet on the side where the sum is $120^{\circ}$ (which is less than $180^{\circ}$).

(C) According to Euclid's geometry,the hierarchy of geometric figures is as follows:
$1$. $A$ solid has three dimensions (length,breadth,and height).
$2$. The boundary of a solid is a surface,which has two dimensions (length and breadth).
$3$. The boundary of a surface is a line,which has one dimension (length).
$4$. The end of a line is a point,which has no dimension.
Therefore,the sequence from solids to points is: Solids $\rightarrow$ Surfaces $\rightarrow$ Lines $\rightarrow$ Points.

(D) solid is a three-dimensional object that has length,breadth,and height. It occupies space and can be moved from one place to another. Therefore,a solid has $3$ dimensions. Examples include a cuboid,cube,cylinder,and cone.

(A) In geometry,a point has $0$ dimensions,a line has $1$ dimension,and a surface has $2$ dimensions (length and breadth). Therefore,the number of dimensions a surface has is $2$.

(B) According to Euclid's geometry,a point is defined as that which has no part.
Since it has no length,no breadth,and no height,it does not occupy any space.
Therefore,a point has $0$ dimensions.

(C) Euclid divided his famous treatise "The Elements" into $13$ chapters,which are known as books.

(D) Euclid's work,titled 'Elements',is a collection of $13$ books.
These books contain a total of $465$ propositions.
Therefore,the correct answer is $465$.

(A) According to Euclid's geometry,the boundaries of solids are surfaces. $A$ solid has three dimensions (length,breadth,and height),and its boundary is a surface,which has two dimensions.

(B) According to Euclid's definitions,a surface is that which has length and breadth only. The edges or boundaries of a surface are curves.

(B) Euclid's second axiom states that if equals are added to equals,the wholes are equal.
In the given statement,we have $x+y=10$. Adding $z$ to both sides,we get $(x+y)+z = 10+z$.
Since $x+y$ and $10$ are equal,adding the same quantity $z$ to both sides maintains equality.
Therefore,this illustrates the second axiom.

(A) The Greeks were primarily interested in establishing the truth of the statements they discovered using deductive reasoning.
Deductive reasoning involves using established axioms,postulates,and previously proven theorems to arrive at a logical conclusion.
$A$ Greek mathematician,Thales,is credited with giving the first known proof using this method.

(C) Euclid belongs to the country Greece.
Euclid,around $300 \, B.C.$,collected all known work in the field of mathematics and arranged it in his famous treatise called Elements.

(D) Pythagoras $(572 \, BC)$ was a student of Thales.
Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent.
This process continued until $300 \, BC$.
At that time,Euclid,a teacher of mathematics at Alexandria in Egypt,collected all the known work and arranged it in his famous treatise.

(A) $Theorem$ is a mathematical statement that has been proven based on previously established statements such as other theorems,axioms,and postulates. Unlike axioms and postulates,which are assumed to be true without proof,a theorem requires a logical proof to be accepted as a valid mathematical truth.

(B) Euclid's fourth postulate states that all right angles are equal to one another. Therefore,the correct answer is a postulate.

(C) In Euclidean geometry,a statement that describes the meaning of a term or concept is known as a definition. The statement 'Lines are parallel if they do not intersect' defines the concept of parallel lines. Therefore,it is a definition.

(TRUE) The statement is True.
The geometry of the Vedic period originated with the construction of 'vedis' (altars) and fireplaces for performing Vedic rites.
Household rituals used altars with square and circular shapes,whereas public worship required altars with complex shapes,which were combinations of rectangles,triangles,and trapeziums.

(TRUE) True. In geometry,to define a point,a line,and a plane,we would need to define many other terms,which would lead to an infinite chain of definitions. To avoid this,mathematicians have agreed to treat these fundamental geometric concepts as undefined terms.

(TRUE) The statement is True.
According to Euclid's first axiom: 'Things which are equal to the same thing are equal to one another.'
Let the area of the triangle be $A_1$,the area of the rectangle be $A_2$,and the area of the square be $A_3$.
Given: $A_1 = A_2$ and $A_2 = A_3$.
Since both $A_1$ and $A_3$ are equal to the same quantity $A_2$,it follows that $A_1 = A_3$.
Therefore,the area of the triangle is equal to the area of the square.

(A) The statement is True.
Euclid's fourth axiom states that things which coincide with one another are equal to one another. This means that if two objects can be placed on top of each other such that they cover each other completely,they are identical in size and shape,and thus equal.

(TRUE) True. Euclidean geometry is based on the assumption of a flat surface (Euclidean plane). It fails on curved surfaces. For example,on a spherical surface,the sum of the angles of a triangle is always greater than $180^{\circ}$.

(B) The given statement is $False$.
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid,which describes the properties of points,lines,and planes in a flat,two-dimensional space (Euclidean plane).
It is specifically designed for flat surfaces. Geometry on curved surfaces is known as non-Euclidean geometry (such as spherical or hyperbolic geometry).
Therefore,Euclidean geometry is not valid for curved surfaces.

(FALSE) The given statement is False.
Justification: According to Euclid's geometry,the boundaries of solids are surfaces,not curves. $A$ solid has three dimensions (length,breadth,and height),its boundaries are surfaces (two-dimensional),and the boundaries of surfaces are curves or straight lines (one-dimensional).

(B) The given statement is False.
According to Euclid's geometry,the edges of a surface are lines,not curves. $A$ surface is defined as that which has length and breadth only,and its boundaries or edges are lines.

(A) True.
This statement is one of Euclid's axioms. Euclid's axioms are fundamental assumptions used in geometry. Specifically,Euclid's sixth axiom states that "Things which are double of the same things are equal to one another."
If we have two quantities $x$ and $y$ such that both are double of a quantity $z$,then $x = 2z$ and $y = 2z$. Since both $x$ and $y$ are equal to $2z$,it follows that $x = y$.

(TRUE) The given statement is True.
According to Euclid's Axiom,"The whole is greater than the part."
Specifically,Euclid's third common notion states that if a quantity $B$ is a part of another quantity $A$,there exists another quantity $C$ such that $A = B + C$.

(FALSE) The given statement is False.
Justification: Axioms are the basic assumptions or postulates that are taken for granted without proof. The statements that are proved using these axioms,postulates,and previously proven statements are called theorems.

(A) The given statement is True.
Justification: Playfair's axiom states that 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 $l$. This axiom is logically equivalent to Euclid's fifth postulate,which is a fundamental principle in Euclidean geometry.

(A) The statement is $True$.
Justification: According to Playfair's Axiom,which is an equivalent version of Euclid's fifth postulate,if a line $l$ and a point $P$ (not on $l$) are given,there exists exactly one line $m$ passing through $P$ such that $m$ is parallel to $l$.
If two distinct intersecting lines $m_1$ and $m_2$ were both parallel to the same line $l$,it would contradict the property that through a point not on a given line,there is only one unique line parallel to the given line. Therefore,two distinct intersecting lines cannot be parallel to the same line.