Introduction to Euclid's Geometry — Question 4
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According to Euclid's axioms, we know that when equals are added to equals, the wholes are equal.
Given : AC = BC
Adding AC on both sides, we get
⇒ AC + AC = BC + AC (BC + AC coincides with AB)
⇒ 2 AC = AB
⇒ AC = AB
⇒ Hence, proved that AC = AB.
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)