Advertisements
Advertisements
Question
Read the following statements which are taken as axioms:
- If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
- If a transversal intersect two parallel lines, then alternate interior angles are equal.
Is this system of axioms consistent? Justify your answer.
Advertisements
Solution
A system of axiom is called consistent, if there is no statement which can be deduced from these axioms such that it contradicts any axiom. We know that, if a transversal intersects two parallel lines, then each pair of corresponding angles are equal, which is a theorem. So, Statement I is false and not an axiom.
Also, we know that, if a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. It is also a theorem. So, Statement II is true and an axiom.
Thus, in given statements, first is false and second is an axiom.
Hence, given system of axioms is not consistent.
APPEARS IN
RELATED QUESTIONS
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
radius of a circle
Give a definition of the following term. Are there other terms that need to be defined first? What are they, and how might you define them?
square
How many lines can be drawn through a given point.
Boundaries of surfaces are ______.
‘Lines are parallel, if they do not intersect’ is stated in the form of ______.
The boundaries of the solids are curves.
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.
Solve the following question using appropriate Euclid’s axiom:
In the following figure, if OX = `1/2` XY, PX = `1/2` XZ and OX = PX, show that XY = XZ.
Read the following axioms:
- Things which are equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent.
The following statement is true or false? Give reason for your answer.
There are an infinite number of lines which pass through two distinct points.
