We are now in possession of a theory of parallels and a theory of perpendiculars and a theory of motion, and from these theories the theory of congruence can be constructed. It will be remembered that a family of parallel levels in any moment is the family of levels in which that moment is intersected by the family of moments of some other time-system. Also a family of parallel moments is the family of moments of some one time-system. Thus we can enlarge our concept of a family of parallel levels so as to include levels in different moments of one time-system. With this enlarged concept we say that a complete family of parallel levels in a time-system α is the complete family of levels in which the moments of α intersect the moments of β. This complete family of parallel levels is also evidently a family lying in the moments of the time-system β. By introducing a third time-system γ, parallel rects are obtained. Also all the points of any one time-system form a family of parallel point-tracks. Thus there are three types of parallelograms in the four-dimensional manifold of event-particles. In parallelograms of the first type the two pairs of parallel sides are both of them pairs of rects. In parallelograms of the second type one pair of parallel sides is a pair of rects and the other pair is a pair of point-tracks. In parallelograms of the third type the two pairs of parallel sides are both of them pairs of point-tracks. The first axiom of congruence is that the opposite sides of any parallelogram are congruent. This axiom enables us to compare the lengths of any two segments either respectively on parallel rects or on the same rect. Also it enables us to compare the lengths of any two segments either respectively on parallel point-tracks or on the same point-track. It follows from this axiom that two objects at rest in any two points of a time-system β are moving with equal velocities in any other time-system α along parallel lines. Thus we can speak of the velocity in α due to the time-system β without specifying any particular point in β. The axiom also enables us to measure time in any time-system; but does not enable us to compare times in different time-systems. The second axiom of congruence concerns parallelograms on congruent bases and between the same parallels, which have also their other pairs of sides parallel. The axiom asserts that the rect joining the two event-particles of intersection of the diagonals is parallel to the rect on which the bases lie. By the aid of this axiom it easily follows that the diagonals of a parallelogram bisect each other.