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. 2 This is the name given to pairs of corresponding points in the two figures.

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