theorems and must be proved by a particular line of reasoning. To produce this
reasoning, we impose on ourselves the conditions of the hypothesis and, assuming
that these conditions are satisﬁed, we must deduce the facts stated in the conclusion.
According to this, we allow that a certain fact is true:
If it is part of the hypothesis;
If it is part of the deﬁnition of one of the elements under
If it follows from an axiom;
4◦. If it follows from an earlier proof.
No fact can be considered certain, in a geometric argument, if it does not follow
from one of the preceding four reasons.
2b. A proposition whose conclusion is formed, completely or partially, by the
hypothesis of a ﬁrst proposition, and whose hypothesis is formed, completely or
partially, by the conclusion of the ﬁrst proposition, is called a converse of the ﬁrst
An immediate consequence of a theorem is called a corollary.
On the other hand, a lemma is a preparatory proposition, intended to facilitate
the proof of a later proposition.
3. Congruent ﬁgures. An arbitrary geometric ﬁgure can be moved through
space without deformation in inﬁnitely many ways, as is the case with ordinary
Two ﬁgures are called congruent if one can be moved onto the other, in such
a way that all their parts
in other words, two congruent ﬁgures are one
and the same ﬁgure, in two diﬀerent places.
A ﬁgure which is subject to displacement without deformation is also called an
4. The Straight Line. The simplest of all lines is a straight line, of which a
stretched thread gives us an idea. The notion of a straight line is clear by itself; in
order to use it in our reasoning, we consider the straight line to be deﬁned by its
obvious properties, in particular the following:
Any ﬁgure congruent to a straight line is itself a straight line; and con-
versely, every straight line can be made to coincide with any other straight line, and
this may be done so that an arbitrarily chosen point on the ﬁrst coincides with any
point chosen on the second;
Through any two points we can draw one, and only one, straight line.
Thus we can talk about the straight line passing through points A and B or,
more briefly, of the straight line AB.
It follows immediately from the deﬁnition that two diﬀerent lines can meet in
only one point, since, if they had two common points, they could not be distinct.
often happens in the course of a proof that one introduces auxiliary elements into the
ﬁgure. A fact can then be true by virtue of the deﬁnition of these new elements. We then say
that the fact is true by construction.
uses the term equal throughout to describe ﬁgures that can be made to coincide.
The term congruent is used more often in English. We will use either term for line segments and
angles (but most often the term equal for these simple ﬁgures), and the term congruent for other,
more complicated ﬁgures.–transl.