- ATransitive
- BSubtractive
- CAdditive
- DReflexive
Step-by-Step Solution
a = b and b = c giving a = c is the transitive property, captured by Euclid's first axiom ('things equal to the same thing are equal to one another').
Answer: A) Transitive
If a, b, c are points such that a = b and b = c, then a = c. The relation of equality here is said to be ____.
Class 9Introduction to Euclid's GeometryGeometry — Foundations
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.
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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