48 VI. ON PARALLELOGRAMS. — ON TRANSLATIONS with equal opposite sides is thus formed by the non-parallel sides and the diagonals of an isosceles trapezoid. Remark III. Two quadrilaterals with four pairs of equal sides need not be congruent. In other words, we can deform a quadrilateral ABCD (proper or not) without changing the lengths of its sides. Indeed, let AB = a, BC = b, CD = c, DA = d be the constant lengths of the four sides. With sides a, b and an arbitrary included angle B, we can construct a triangle ABC, whose construction will determine the length of AC (a diagonal of the quadrilateral). To each value of this diagonal (under the conditions for existence given in 86, Book II) there corresponds a triangle ACD with this base AC and with the other two sides CD, DA having length c, d. The angle B can therefore be arbitrary (at least within certain limits). A quadrilateral which can be deformed under these conditions is called an artic- ulated quadrilateral. This notion is important in practical applications of geometry. According to the preceding results, a parallelogram remains a parallelogram when it is articulated and, likewise, an anti-parallelogram remains an anti-parallelo- gram under these conditions.1 47. Theorem. In a parallelogram, the two diagonals divide each other into equal parts. In parallelogram ABCD (Fig. 46), we draw diagonals AC, BD, which intersect at O. Triangles ABO, CDO are congruent because they have equal angles, and an equal side AB = CD (by the preceding theorem). Therefore AO = CO, BO = DO. QED Converse. A quadrilateral is a parallelogram if its diagonals divide each other into equal parts. Figure 46 Indeed, assume (Fig. 46) that AO = CO, BO = DO. Triangles ABO, CDO are again congruent because they have an equal angle (the angles at O are vertical) included between pairs of equal sides. Therefore their angles at A and C will be equal, so that AB is parallel to CD. The congruence of triangles ADO, BCO also shows that AD is parallel to BC. 1 This reasoning fails only if a parallelogram is transformed into an anti-parallelogram, or inversely (since a parallelogram and an anti-parallelogram are the only quadrilaterals with opposite sides equal). If the deformation occurs continuously, this requires that the quadrilateral first flatten into a line, so that one pair of adjacent sides ends up as extensions of each other, as does the other pair.

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