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If , then because equals were subtracted from equals. Which axiom is used?

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

Step-by-Step Solution

Subtracting 10 from both sides uses the axiom: equals subtracted from equals give equal remainders.
Answer: A) If equals are subtracted from equals, the remainders are equal

Key idea: Euclid's axioms (common notions)

Axioms apply to all of mathematics, not just geometry. The key ones:

1. Things equal to the same thing are equal to one another. (If a = c and b = c, then a = b.)
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

These feel obvious — that's the point. Euclid wanted his foundation to be impossible to argue with.

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