Tag: introduction to euclid's geometry

Questions Related to introduction to euclid's geometry

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.

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

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.

The sum of angles of a triangle is _____

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

  1. $180^\circ$

  2. $135^\circ$

  3. $90^\circ$

  4. $45^\circ$

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Correct Option: A
Explanation:

Sum of angles is given by the formula $(n-2) \times 180$,  where $n=no. of sides$

So, for triangle $n=3$, hence sum of angles$=180^o$

How many parallel lines can be drawn to a given line

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

  1. $0$

  2. $1$

  3. $2$

  4. Infinite

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Correct Option: D
Explanation:

Infinite lines can be drawn parallel to a given line from 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
maths introduction to euclid's geometry euclid's fifth postulate conditional statements and converse euclid's postulates

  1. Fifth Postulate

  2. First Postulate

  3. Second Postulate

  4. Third Postulate

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Correct Option: A
Explanation:

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

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.

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

  1. True

  2. False

Show answer
Correct Option: B
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.
$B$

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

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

  1. Solids - surfaces - lines - points

  2. Solids - lines - surfaces - points

  3. Solids - surfaces - element - points

  4. Solids - elements - surfaces - points

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Correct Option: A
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

Two distinct ________ lines cannot be parallel to the same line.

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

  1. Intersecting

  2. Non-intersecting

  3. Parallel

  4. None of these

Show answer
Correct Option: A
Explanation:

Two intersecting lines cannot be parallel to same line as this statement is equivalent to Euclid's fifth postulate.