Easy
Read the following statements which are taken as axioms:
$(i)$ If a transversal intersects two parallel lines,then corresponding angles are not necessarily equal.
$(ii)$ If a transversal intersects two parallel lines,then alternate interior angles are equal.
Is this system of axioms consistent? Justify your answer.
$(i)$ If a transversal intersects two parallel lines,then corresponding angles are not necessarily equal.
$(ii)$ If a transversal intersects two parallel lines,then alternate interior angles are equal.
Is this system of axioms consistent? Justify your answer.
(N/A) No,this system of axioms is not consistent.
In Euclidean geometry,if a transversal intersects two parallel lines,the corresponding angles must be equal.
If we assume statement $(i)$ is true,it contradicts the fundamental property of parallel lines.
Furthermore,if corresponding angles are not equal,then the alternate interior angles cannot be equal,which contradicts statement $(ii)$.
Since the two statements lead to a logical contradiction,the system of axioms is inconsistent.
In Euclidean geometry,if a transversal intersects two parallel lines,the corresponding angles must be equal.
If we assume statement $(i)$ is true,it contradicts the fundamental property of parallel lines.
Furthermore,if corresponding angles are not equal,then the alternate interior angles cannot be equal,which contradicts statement $(ii)$.
Since the two statements lead to a logical contradiction,the system of axioms is inconsistent.
Explore More
Similar Questions
Write whether the following statement is True or False. Justify your answer.
If a quantity $B$ is a part of another quantity $A$,then $A$ can be written as the sum of $B$ and some third quantity $C$.
If a quantity $B$ is a part of another quantity $A$,then $A$ can be written as the sum of $B$ and some third quantity $C$.
Easy
View SolutionState whether the following statement is True or False. Justify your answer.
"For every line $l$ and for every point $P$ not lying on a given line $l$,there exists a unique line $m$ passing through $P$ and parallel to $l$." This is known as Playfair's axiom.
"For every line $l$ and for every point $P$ not lying on a given line $l$,there exists a unique line $m$ passing through $P$ and parallel to $l$." This is known as Playfair's axiom.
Easy
View SolutionIn the Indus Valley Civilisation (about $3000 \, BC$),the bricks used had the ratio of $\ldots \ldots$ for length,breadth,and height,respectively.
Easy
View SolutionSolve the following question using appropriate Euclid's axiom:
In the figure:
$BM = BN$,$M$ is the mid-point of $AB$ and $N$ is the mid-point of $BC$. Show that $AB = BC$.
In the figure:
$BM = BN$,$M$ is the mid-point of $AB$ and $N$ is the mid-point of $BC$. Show that $AB = BC$.
Medium
View SolutionState whether the following statement is true or false:
Euclid was a well-known student of Thales.
Euclid was a well-known student of Thales.
Easy
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo