- AAB = XY
- BAB > XY
- CAB < XY
- DCannot be concluded
Step-by-Step Solution
Axiom 1: things equal to the same thing are equal to one another. AB and XY both equal PQ, so AB = XY.
Answer: A) AB = XY
If AB = PQ and PQ = XY, by Euclid's axiom we can conclude that ____.
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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