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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