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C is a point on segment AB with AC = 5 cm (orange) and CB = 3 cm (blue), as shown. Which statement follows from Euclid's axiom 'the whole is greater than the part'?

Class 9Introduction to Euclid's GeometryGeometry — Foundations
  • AAB = AC, because C lies on AB
  • BAB > AC, because AB = AC + CB = 8 cm
  • CAC > AB, because AC is the longer piece
  • DAB < AC + CB, because a whole is less than its parts together

Step-by-Step Solution

is the whole and is a part of it: cm, so (and also ).
Answer: B) AB > AC, because AB = AC + CB = 8 cm

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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