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If AB = CD, can we add a common segment BE to both to conclude AE = CD + BE only when E extends correctly? Simpler: if AB = CD and we add BC = BC to both, which axiom gives AB + BC = CD + BC?

Class 9Introduction to Euclid's GeometryGeometry — Foundations
  • AIf equals are added to equals, the wholes are equal
  • BIf equals are subtracted from equals, remainders are equal
  • CThe whole is greater than the part
  • DThings which coincide are equal

Step-by-Step Solution

Adding the equal quantity BC to the equal quantities AB and CD keeps the sums equal, by the 'equals added to equals' axiom.
Answer: A) If equals are added to equals, the wholes are equal

Key idea: Axiom vs Postulate — the difference

Both are starting assumptions, accepted without proof. The difference is **scope**:

- **Axiom**: a universal truth used across all of mathematics (arithmetic, algebra, geometry).
- **Postulate**: a starting assumption specific to one subject — for Euclid, geometry.

Modern mathematicians often blur this distinction and use 'axiom' for both, but at this level you must keep them separate.

Example:
- 'If equals are added to equals, the wholes are equal' → AXIOM (general arithmetic).
- 'A circle can be drawn with any centre and radius' → POSTULATE (only about geometry).

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