Read the following axioms:
$(i)$ Things which are equal to the same thing are equal to one another.
$(ii)$ If equals are added to equals,the wholes are equal.
$(iii)$ Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent.
$(i)$ Things which are equal to the same thing are equal to one another.
$(ii)$ If equals are added to equals,the wholes are equal.
$(iii)$ Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent.
(A) The given system of axioms is consistent.
$(i)$ This is Euclid's first axiom: 'Things which are equal to the same thing are equal to one another.'
$(ii)$ This is Euclid's second axiom: 'If equals are added to equals,the wholes are equal.'
$(iii)$ This is Euclid's third axiom: 'Things which are double of the same thing are equal to one another.'
Since all the given statements are standard axioms defined by Euclid,they do not contradict each other and are therefore consistent.
$(i)$ This is Euclid's first axiom: 'Things which are equal to the same thing are equal to one another.'
$(ii)$ This is Euclid's second axiom: 'If equals are added to equals,the wholes are equal.'
$(iii)$ This is Euclid's third axiom: 'Things which are double of the same thing are equal to one another.'
Since all the given statements are standard axioms defined by Euclid,they do not contradict each other and are therefore consistent.
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Solve the following question using an appropriate Euclid's axiom:
In the figure,we have $AC = DC$ and $CB = CE$. Show that $AB = DE$.
In the figure,we have $AC = DC$ and $CB = CE$. Show that $AB = DE$.
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View SolutionRead the following statement:
An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are $60^{\circ}$ each.
Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?
An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are $60^{\circ}$ each.
Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?
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View SolutionState whether the following statement is true or false:
Two distinct lines cannot have more than one point in common.
Two distinct lines cannot have more than one point in common.
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View SolutionEuclid was a native of $\ldots \ldots \ldots .$
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