50 VI. ON PARALLELOGRAMS. — ON TRANSLATIONS
Figure 48
49. A square is a quadrilateral in which all sides are equal, and all the angles
right angles.
Thus a square is both a rhombus and a rectangle, so that the diagonals are
equal, perpendicular, and intersect each other at their common midpoint.
Conversely, any quadrilateral whose diagonals are equal, perpendicular, and
bisect each other is a square.
Two squares with the same side are congruent.
50. Translations.
Lemma. Two figures F , F are congruent, with the same sense of rotation, if
their points correspond in such a way that, if we take points A, B, C from one
figure, and the corresponding points A , B , C from the other, then the triangles
thus formed are always congruent, with the same sense of rotation, no matter which
point we take for C.
Indeed, take two points A, B in figure F , and the
homologous2
points A , B .
Segment AB is then clearly equal to A B . Let us move the second figure onto the
first in such a way that these two equal segments coincide. We claim that then
the two figures coincide completely. Indeed, let C be a third point of the first
figure, and let C be its homologous point. Since the two triangles ABC, A B C
are congruent, angle B A C is equal to BAC, and has the same sense of rotation.
Therefore, when A B is made to coincide with AB, the line A C will assume the
direction of AC. Since A C = AC as well, we conclude that C coincides with
C. This argument applies to all the points of the figure, so the two figures must
coincide completely.
Remarks. I. We have just provided a sufficient condition for two figures to be
congruent; this conditions is also clearly necessary.
From the preceding reasoning it also follows that:
II. In order to superimpose two equal figures with the same orientation, it
suffices to superimpose two points of one of the figures onto their homologous
points.
2This
is the name given to pairs of corresponding points in the two figures.
