Tag: euclid's fifth postulate

Questions Related to euclid's fifth postulate

The mathematical statements that are proved are called axioms.

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

  1. True

  2. False

  3. cannot be determined

  4. None of the above

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Correct Option: B
Explanation:

False, statements that are proved are theorems.

A proof is required for :

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

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition

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Correct Option: C
Explanation:

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


Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In a mathematical paper, the term theorem is often reserved for the most important results.
  
So, the correct option is $C$ as a theorem needs a proof.

A lemma is a proven statement used for proving another statement.

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

  1. True

  2. False

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Correct Option: A
Explanation:

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

A theorem is:

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

  1. an assumption

  2. always true

  3. always false

  4. sometimes true and sometimes false

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Correct Option: B
Explanation:

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.


Example: In a Euclidean space, the sum of measures of the three angles of any triangles is invariably equal to the straight angle, also as $180^o$

According to Euclid, a surface has ____.

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

  1. Length but no breadth and thickness

  2. Length and breadth but no thickness

  3. No length, no breadth and no thickness

  4. Length, breadth and thickness

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Correct Option: B
Explanation:

According to Euclid a surface is a two-dimension plane without any volume, hence it has length and breath but no thickness.

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

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

  1. an axiom

  2. a definition

  3. a postulate

  4. a proof

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Correct Option: C
Explanation:

Euclid's fourth Postulate states that all right angles are equal to each other.

Ans- Option C.

Use Euclid's division algorithm to find the HCF of 
$867$ and $255$

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

  1. $50$

  2. $51$

  3. $41$

  4. $3$

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Correct Option: B

___________ was the most logical and abstract creator of Euclid's geometry approach.

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

  1. Hilbert

  2. Bhasharacharya

  3. Thelus

  4. Pythagorous

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Correct Option: D
Explanation:

Pythagoras developed the theory of geometry to a great extent.

STATEMENT -1 : Given positive integers a and b, there exist whole numbers q and r satisfying a $=$ bq + r, 0 $\leq$ r  < b.
STATEMENT -2 : Any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

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

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True ; Statement - 1 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True

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Correct Option: B
Explanation:

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

Consider the following statement: 

There exists a pair of straight lines that are everywhere equidistant from one another. 
Is this statement a direct consequence of Euclid's fifth postulate? Explain

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

  1. True

  2. False

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Correct Option: A
Explanation:

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.