Easy
Read the following two statements which are taken as axioms:
$(i)$ If two lines intersect each other,then the vertically opposite angles are not equal.
$(ii)$ If a ray stands on a line,then the sum of two adjacent angles so formed is equal to $180^{\circ}$.
Is this system of axioms consistent? Justify your answer.
$(i)$ If two lines intersect each other,then the vertically opposite angles are not equal.
$(ii)$ If a ray stands on a line,then the sum of two adjacent angles so formed is equal to $180^{\circ}$.
Is this system of axioms consistent? Justify your answer.
(B) The given system of axioms is not consistent.
Statement $(ii)$ is a standard geometric property (Linear Pair Axiom) which is universally true in Euclidean geometry.
If we accept statement $(ii)$,then for two intersecting lines,the angles formed on a straight line must sum to $180^{\circ}$.
By applying this property to both sides of the intersection,it can be mathematically proven that vertically opposite angles are always equal.
Since statement $(i)$ contradicts this proven result derived from statement $(ii)$,the system of axioms is inconsistent.
Statement $(ii)$ is a standard geometric property (Linear Pair Axiom) which is universally true in Euclidean geometry.
If we accept statement $(ii)$,then for two intersecting lines,the angles formed on a straight line must sum to $180^{\circ}$.
By applying this property to both sides of the intersection,it can be mathematically proven that vertically opposite angles are always equal.
Since statement $(i)$ contradicts this proven result derived from statement $(ii)$,the system of axioms is inconsistent.
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Similar Questions
Read 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?
Medium
View SolutionIn the Indus Valley Civilisation (about $3000\, B.C.$),the bricks used for construction work had dimensions in the ratio:
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View SolutionBoundaries of solids are:
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View SolutionRead 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.
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View SolutionState whether the following statement is True or False. Justify your answer.
Euclidean geometry is valid only for curved surfaces.
Euclidean geometry is valid only for curved surfaces.
Easy
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