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If , then . Which of Euclid's axioms justifies this conclusion?

Class 9Introduction to Euclid's GeometryGeometry — Foundations
  • AThings which are halves of the same things are equal
  • BThings which are doubles of the same things are equal
  • CThe whole is greater than the part
  • DThings which coincide are equal

Step-by-Step Solution

a and are the halves of the equal quantities and , so by the 'halves of equals' axiom.
Answer: A) Things which are halves of the same things 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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