Tag: axioms, postulates and theorems

Questions Related to axioms, postulates and theorems

Which of the following is Euclid's first postulate?

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

  1. All right angles are equal to one another.

  2. The whole is greater then the part.

  3. A circle can be drawn with any centre and any radius.

  4. A straight line segment can be drawn joining any two points.

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Correct Option: D
Explanation:

Euclid's first postulate is :

A straight line segment can be drawn joining any two points.
Thus, option D is correct.

If point $P$ lies on $AB$, then $AB$ is always greater than $AP$. This concept is on which of the following Euclid's Axioms.

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

  1. First Axiom

  2. Second Axiom

  3. Third Axoim

  4. Fifth Axiom

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Correct Option: D
Explanation:

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

Here AB is the whole line and AP is the part and according to the fifth axiom we have AB is always greater than AP.
So the given statement is Euclid's fifth axiom.

State whether the following statement is true/false

Use Euclid's division lemma we can show that the square of any positive integer is either of the form $3m$ or $3m+1$ for some integer m.

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

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Correct Option: A

Axioms are assumed

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

  1. universal truths in all branches of mathematics

  2. universal truths specific to geometry

  3. theorems

  4. definitions

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Correct Option: A
Explanation:

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

No mathematical deduction is needed to prove them.
Practically they are starting points of reasoning.

So axioms are assumed universal truths in all branches of mathematics.

Ans- Option A.

John is of the same age as Mohan. Ram is also of the same age as Mohan. State the Euclid's axiom that illustrates the relative ages of John and Ram

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

  1. First axiom

  2. Second axiom

  3. Third axiom

  4. Fourth axiom

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Correct Option: A
Explanation:
Given that
John's age=Mohan's age &
Ram's age=Mohan's age.
So, by the first axiom of Euclid, 
John's age=Ram's age.
Euclid's first axiom states that
things, which are equal to the same thing, are equal to one another.

Ans- Option A.

$\angle A=\angle B$ and $\angle B=\angle C$, According to which axiom of Euclid the relation between $\angle A$ and $\angle C$ is established?

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

  1. I

  2. II

  3. III

  4. IV

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Correct Option: A
Explanation:

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

Then, according to Euclid's first axiom, which states that 
"things which are   equal to the same thing are also equal to each other",
 $\quad \angle A=\angle C\quad $
Ans- Option A.

Euclid's fourth axiom says that everything equals itself.

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. Ambiguous

  4. Data insufficient

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Correct Option: A
Explanation:

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

So the given statement is true by the axiom IV.
Ans- Option A.

The boundaries of the solids are called curves.

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. Ambiguous

  4. Data Insufficient

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Correct Option: B
Explanation:

The boundaries of the solids are called surfaces.

While the boundaries of the surfaces are called curves.

The Euclidean geometry is valid only for figures in the plane.

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. Ambiguous

  4. Data Insufficient

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Correct Option: A
Explanation:

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.  

So, in space or in multidimensional space the Euclidean axioms are not applicable.
Ans- Option A.

According to Euclid : The whole is greater than the part .State whether that this is true or false.
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

Show answer
Correct Option: A
Explanation:

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