Meena lays stick P beside stick Q and the ends match (Step 1). Then she lays stick R beside the same stick Q and the ends match again (Step 2). Without ever putting P beside R, she says P = R. Which of Euclid's axioms is she using?

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.