Tag: euclid's fifth postulate

Questions Related to euclid's fifth postulate

Euclidean geometry is valid only for curved surfaces.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. True

  2. False

  3. Sometimes True

  4. Data Insufficient

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Correct Option: B
Explanation:

Euclid's postulates:

$\rightarrow$A straight line can be drawn joining any two points.
$\rightarrow$A straight line segment can be extended indefinitely in a straight line.
$\rightarrow$A circle can be drawn having segment as radius and one endpoint as center. 
$\rightarrow$All right angles are congruent and equal.
$\rightarrow$Parallel postulate.

According to Euclid's axioms, the _____ is greater than the part.

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Half

  2. Large

  3. Whole

  4. None of these

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Correct Option: C
Explanation:

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


Proof:
Let's take whole $3$ and part $\dfrac{1}{3}$
Subtracting, we get
$3-\dfrac{1}{3} = \dfrac{8}{3} > 0$

Hence, Proved that whole is greater than part.

Two intersecting lines cannot be parallel to the same line is stated in the form of :

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. an axiom

  2. a definition

  3. a postulate

  4. a proof

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Correct Option: A
Explanation:

Axiom

example if line A and line B are intersecting and line C is parallel to line A then line C is not parallel to line B.
$A$

Using Euclid's Division Lemma, for any positive integer $n, n^3-n$ is always divisible by 

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $6$

  2. $4$

  3. $3$

  4. $8$

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Correct Option: A
Explanation:

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


[$\because$ if $P=ab+r$, Then $0\le r < a$ by Euclid lemma]


$\therefore$ Let $n=3r, 3r+1,3r+2$ where $r$ is an integer

Case $1$: When $n=3r$

Then, $n^{3}-n$ is divisible by $3$  [$\because n^{3}-n=n(n-1)(n+1)=3r(3r-1)(3r+1)$, early shown it is divisible by $3$]

Case $2$: When $n=3r+1$

$n-1=3r+1-1=3r$

Then, $n^{3}-n=(3r+1)(3r)(3r+2)$ it is divisible by $3$

Case: when $n=3r-1$

$m+1=3r-1+1=3r$

Then, $n^{3}-n=(3r-1)(3r-2)(3r)$ it is divisible by $3$

Now out of three $(n-1)^{n}$ and $(n+1)$ are must be even so it is divisible by $2$

$n^{3}-n$ is divisible by $2\times 3=6$

Euclid stated that if equals are subtracted from equals, the remainders are equals in the form of :

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. an axiom

  2. a postulate

  3. a definition

  4. a proof

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Correct Option: A
Explanation:

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

The things which coincide with one another are:

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. equal to another

  2. unequal

  3. double of same thing

  4. Triple of same things

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Correct Option: A
Explanation:

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

Hence, $A$ is correct.

Euclid's stated that all right angles are equal to each other in the form of :

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. an axiom

  2. a definition

  3. a postulate

  4. a proof

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Correct Option: C
Explanation:

One of Euclid's five postulates is:

$All$ $right$ $angles$ $are$ $CONGRUENT$.
So, the correct option is  $C$.

Which of the following is Euler's formula?

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. $F+V=E+2$

  2. $F+E=V+2$

  3. $F+E-V=2$

  4. $F+2=E+V$

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Correct Option: A
Explanation:

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


Option $A$ is the correct answer.

Euclid stated that all right angles are equal to one another in the form of a/an ..........

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. Axiom

  2. Defination

  3. Postulate

  4. Proof

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Correct Option: C
Explanation:

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 is

maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

  1. the 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

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Correct Option: B
Explanation:

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.