Textbook - Introduction to Euclid’s Geometry 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.