the axioms of congruence
Once you can say which point lies on which line and which point is between two others, you still cannot say two segments are 'the same length' or two angles are 'the same size' — incidence and order know nothing about magnitude. The axioms of congruence are the family that supplies this. They make precise, without ever mentioning a number or a ruler, what it means for one segment to be copyable onto another, and for shapes to match.
Hilbert states them for segments and for angles. For segments: given a segment AB and a ray starting at a point C, there is exactly one point D on that ray with CD congruent to AB — you can lay off a copy of any segment, uniquely, in any direction. Congruence of segments is also required to be transitive (if AB is congruent to CD and CD to EF, then AB to EF) and additive (congruent pieces add to congruent wholes). For angles there is a matching copying axiom: any angle can be reproduced, uniquely, on either side of a given ray. The keystone is the SAS axiom: if two sides and the included angle of one triangle are congruent to those of another, then the triangles are congruent throughout. Hilbert takes SAS as an axiom precisely because Euclid's 'proof' of it secretly slid one triangle on top of the other — a motion Euclid never justified.
From this family flow all the triangle-congruence theorems (ASA, SSS, AAS, the isosceles-triangle theorem) and the whole theory of length and angle measure. The honest subtlety: congruence is the geometric notion of 'equal in size and shape', and it is logically prior to numerical measurement — Hilbert builds it first, then shows numbers can be attached afterward. So 'congruent' is not defined as 'has equal measure'; rather, measure is defined so that congruent things get equal numbers.
The copying axiom in action: take segment AB of any length, point C, and a ray from C. The axiom hands you one and only one point D on that ray with CD congruent to AB. That is the logical backbone of 'copying a segment' with compass and straightedge — the construction works because the axiom promises D exists and is unique.
Copy a segment, copy an angle, and SAS — the three pillars that let shapes be compared.
Hilbert makes SAS an axiom, not a theorem, because Euclid 'proved' it by sliding triangles together — an unstated motion. SSS, ASA and AAS then become provable theorems; SSA and AAA still are not valid criteria.