Introduction to Euclid's Geometry — Question 3
Back to all questionsConsider two ‘postulates’ given below:
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
Using Euclid’s axioms to check these postulates :
Yes, these postulates contain undefined terms like point and line.
These two statements are consistent as they talk about two different situations meaning different things.
These statements do not follow Euclid’s postulates but one of the axioms about “Given any two points, a unique line that passes through them” is followed.
Euclid’s Geometry — Interactive Study Guide
The Big Idea
Euclid started with things everyone agrees on (axioms and postulates) and built all of geometry by logical deduction. This is the axiomatic method — start from accepted truths, derive everything else.
Axiom vs Postulate vs Theorem
Axiom: Self-evident truth used everywhere in mathematics.
Postulate: Assumption specific to geometry (cannot be proved, must be accepted).
Theorem: A statement that has been proved using axioms, postulates, and logic.
The Famous 5th Postulate
Euclid’s 5th postulate is about parallel lines. For over 2000 years, mathematicians tried to prove it from the other 4 postulates. They failed — because it’s independent! Changing the 5th postulate leads to non-Euclidean geometry.
Quick Self-Check
- State Euclid’s first axiom. (Things equal to the same thing are equal to one another.)
- How many lines can be drawn through two distinct points? (Exactly one — Postulate 1)
- What is the modern equivalent of Euclid’s 5th postulate? (Playfair’s axiom)