Axioms, postulates and theorems - class-IX

Euclid's first postulate is :

The fifth axiom of Euclid's about geometry is the whole of anything is greater than the part of it.

From time immemorial, axioms have been acquired by man through the day to day experiences .

Given that $\quad \angle A=\angle B\quad & \quad \angle B=\angle C.\quad $

Euclid's fourth axiom states that "things
which coincide with one another are equal to one another."

The boundaries of the solids are called surfaces.

The given statement is true. Because,  by Einstein's  theory of  general  relativity, physical space itself is not Euclidean. Euclidean space is a good approximation  for it where the gravitational field is weak.  

This is Euclid's fifth axiom. Hence $true$.

Euclid belongs to Greece.

The edges of a surface are called curves.

Let each salesman make Rs. $x$ in August.
In September, the sale of each salesman is Rs. $2x$.
According to Euclids sixth axiom, things which are double of the same thing are equal to one another, the wholes are equal.
So, the sales of each salesman are equal.

Let a=2x   and  b=2x  when a, b & x are arbritary numbers or things. 

Axioms are statements which are self evident and are accepted without any proof.

According to the first axiom of Euclid " if equals are added to equals the wholes are equal."

(i) This statement is  Euclid's first Axiom.

If a ray stands on a line, then the sum of two adjacent angles so formed is 180.Conversely if the sum of two adjacent angles is 180, then a ray stands on a line (i.e., the non-common arms form a line).

Hence the above statement is True

According to Euclid's 1st axiom,- Things which are equal to the same thing are also equal to one another

True,

Infinite lines can pass through a given point.

According to Euclid's Axioms, For every two points, $A,,B$ there exists no more than one line that contains each of the points $A,B$.

Let $A$ and $B$ both be equal to $C$

According to the first Euclid's axioms ''Things which are equal to the same thing are equal to one another''.

Solution:

Euclid's postulates:

According to axiom $5$ of Euclid, whole is greater than the part and it is a universal truth.

Axiom

$n^{3}-n=n(n^{2}-1)=n(n-1)(n+1)$ is divisible by $3$ then  possible remainder is $0, 1$ and $2$

The above statement is Euclid's third axiom. So, $A$ is correct.

According to Euclid's postulates, $equal$ things coincide with each other.

One of Euclid's five postulates is:

Euler's formula $F + V = E + 2.$

Postulates


1. A straight line may be drawn from any point to any other point.


2. A terminated line (line segment) can be produced indefinitely.
 3. A circle may be described with any centre and any radius.


4. All right angles are equal to one another.


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 the two straight lines if

produced indefinitely, meet on that side on which the sum of angles is taken together

less than two right angles.

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.
Answer (C) Postulate

Euclid's second axiom can be stated as any terminated straight line can be projected indefinitely or it can be stated as if equals be added to equals, the wholes are equal.

Postulates 
1. A straight line may be drawn from any point to any other point. 
2. A terminated line (line segment) can be produced indefinitely.
 3. A circle may be described with any centre and any radius. 
4. All right angles are equal to one another. 
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
So, (D) option is correct.
Answer (D) Postulate I  A straight line may be drawn from any one point to any other point.

A postulate is a statement that is accepted without proof. Axiom is another name for a postulate. 

Following statement is not a Euclid's postulate:-

Two lines can intersect at one point

This set of axiom is consistent.
(i) If a=b and b=c then a =c
(ii) If a = b and c = d then a+ b =c +d
(iii) If x = 2y and z = 2y then x = z

Axiom, Postulate and definition are self evident & do not need any proof.

Here we are given two lines l and m. We need to prove that they have only one point in common.For the time being, let us suppose that the two lines intersect in two distinct points, say P and Q. So, you have two lines passing through two distinct points P and Q. But this assumption clashes with the axiom that only one line can pass through two distinct points. So, the assumption that we started with, that two lines can pass through two distinct points is wrong. From this, what can we conclude? We are forced to conclude that two distinct lines cannot have more than one point in common.

False, statements that are proved are theorems.

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. 

'Lemma ' is a subsidiary or intermediate theorem in an argument or proof.

A theorem is a Statement (logic) that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements.

A theorem is based on assumption that theorem is true.

Both the statements are true but statement - 2 is not a correct explanation for statement - 1.

Take any line $l$ and a point $P$ not on $l$. Then by play Fair's axiom, which is equivalent to the fifth postulate, we know that there is a unique line m through $P$ which is parallel to $l$.
Now, the distance of a point from a line is the length of the perpendicular from the point to the line. This distance will be the same for any point on $m$ from $l$ and any point on $l$ from $m$. Thus these two lines are everywhere equidistance from one another.

Yes, Euclid's fifth postulate is valid for parallelism of lines because, if a straight line $l$ falls on two straight lines $m$ and $n$ such that sum of the interior angles on one side of $l$ is two right angles, then by Euclid's fifth postulate the line will not meet on this side of $l$. 

Sum of angles is given by the formula $(n-2) \times 180$,  where $n=no. of sides$

Infinite lines can be drawn parallel to a given line from Euclid's Postulates.

Attempts to prove Euclid's Fifth Postulate using other postulates and axioms led to the discovery of several others geometries

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles

Euclid's Consider the three steps from solids to points (solids-surfaces-lines-points). In

each step we lose one extension, also called a dimension. So, a solid has three

dimensions, a surface has two, a line has one and a point has none. Euclid

summarized these statements as definitions.
Answer (A) Solids - surfaces - lines - points

Two intersecting lines cannot be parallel to same line as this statement is equivalent to Euclid's fifth postulate.

$B$ is called the  Triangle theorem.  $A$ can be proved using this theorem, as in an equilateral triangle, all sides are equal and the angles opposite to these congruent sides are Equal. So, $A$ is the corollary of $B$

A result derived from a theorem is its corollary.

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. 

The formation or expression of an opinion or theory without sufficient evidence for proof is known as conjecture.