Introduction to Euclid's Geometry — Question 1
Back to all questionsWhich of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In figure, if AB = PQ and PQ = XY, then AB = XY.
(i) False
Reason — There are infinite no. of lines that passes through a single point.
(ii) False
Reason — By Euclid's axiom : Given two distinct points, there is a unique line that passes through them.
(iii) True
Reason — By Euclid's postulate : A terminated line can be produced indefinitely on both side.
(iv) True
Reason — If two circle are equal then there radii are equal, because if two circle are equal then on superimposing them their center and boundaries coincides and inscribe equal area thus their radii are equal.
(v) True
Reason — According to Euclid's First Axiom,"Things which are equal to the same thing are equal to one another".
Since, AB = PQ and PQ = XY
∴ AB = XY.
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)