A line which is composed of portions of straight lines is called a broken line.
Other lines, which are neither straight nor broken, are called curves.
5. The part of a straight line contained between two points A and B is called
a segment of the line AB, or the distance AB. We can also consider the portion
of a line which is not limited in one direction, but limited in the other by a point.
This is called a ray.
Figure 1 Figure 2
It follows from the remarks above that two arbitrary rays are congruent.
The distance AB is said to be equal to the distance A B if the ﬁrst segment
can be moved onto the second in such a way that A falls onto A and B onto B .
After such a move, it follows from the two properties which deﬁne the notion
of a straight line that the two segments will coincide entirely. Consequently, the
deﬁnition of equal segments agrees with the deﬁnition of congruent ﬁgures given
Two equal segments AB, A B can be made to coincide in two diﬀerent ways;
namely, the point A falling on A and B on B , or A falling on B and B on A . In
other words, one can turn the segment AB around in such a way that each of the
points A, B takes the place of the other.
When two segments AB, BC are on the same line so that one of them is the
extension of the other (Fig. 1), the segment AC is called the sum of the ﬁrst two
segments. The sum of two, and therefore of many, segments is independent of the
order of the summands.9
In order to compare segments, we move them onto the same line, so that they
start from the same point and are oriented in the same direction: for example AB
and AC (Figures 1 and 2). If the points are in the order A, B, C (Fig. 1), the
segment AC is the sum of AB and another segment BC; in this case AC is greater
than AB; if on the contrary, the order is A, C, B, the segment AC is smaller
than AB. In either case, the third segment BC which, when added to one of the
ﬁrst produces the other, is called the diﬀerence of the two segments. Finally, the
points B and C may coincide, in which case we know that the two segments under
consideration are equal.
On each line segment AB, there exists a point M , the midpoint of AB, equally
distant from A and B; every point on the line between A and M is clearly closer
to A than to B, and the opposite is true for points between M and B.
More generally, a line segment can be divided into any number of equal
6. The Plane. An inﬁnite surface such that a line joining any two points on
it is contained entirely on the surface is called a plane.
We will assume that there is a plane passing through any three points in space.
A straight line drawn in a plane separates the surface into two regions called half-
planes, situated on diﬀerent sides of the line. One cannot travel along a continuous
two segments this follows immediately from the preceding paragraph.
mean by this that there exist points on AB which divide this segment into equal parts.
The question as to whether these points can actually be determined with the help of available
instruments will be considered later (Book III).