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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.

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

Does Euclid' fifth postulate imply the existence of parallel lines? Explain

  1. True

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

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 $l$. 

Next, you know that the sum of the interior angles on the other side of line $l$ will also be two right angles. 
Therefore, they will not meet on the other side also. So, the lines $m$ and $n$ never meet and are, therefore, parallel.

Multiple choice maths introduction to euclid's geometry euclid's fifth postulate conditional statements and converse euclid's postulates
Which Euclid's postulate led to the discovery of several other geometries while attempting to prove it using other postulates and axioms

  1. Fifth Postulate

  2. First Postulate

  3. Second Postulate

  4. Third Postulate
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

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

State true or false:

Attempts to prove Euclid's fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

  1. True

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

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

this the fifth postulate,many tried to prove it but at the end they had to assume something which was very closely related to the fifth postulate,they didnot form any new geometries but from where they started they ended at the same point.
$B$

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

Select the correct answer. The three steps from solids to point are

  1. Solids - surfaces - lines - points

  2. Solids - lines - surfaces - points

  3. Solids - surfaces - element - points

  4. Solids - elements - surfaces - points
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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