Tag: euclid's fifth postulate

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

A proof is required for :

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

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

A theorem is:

  1. an assumption

  2. always true

  3. always false

  4. sometimes true and sometimes false
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $180^o$

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

According to Euclid, a surface has ____.

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

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

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.

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

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

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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.