46 VI. ON PARALLELOGRAMS. ON TRANSLATIONS Figure 44 In this theorem, the hypothesis consists of two parts: 1◦. two sides are parallel 2◦. the other two sides are also parallel. The conclusion also has two parts: 1◦. two opposite sides are equal 2◦. the other two sides are also equal. Since we can form a converse by using either the whole or a part of the original conclusion, and vice versa, this theorem has two converses. Converses. A quadrilateral is a parallelogram: 1◦. If the opposite sides are equal 2◦. If two opposite sides are parallel and equal. 1◦. Suppose that AB = CD and AD = BC in quadrilateral ABCD (Fig. 44). We again draw diagonal AC. Triangles ABC, CDA will be congruent, since they have three sides equal in pairs. Thus angles A1 and C1 are equal, and since these are alternate interior angles with respect to transversal AC, lines AB, CD must be parallel. Likewise, the equality of A2, C2 proves that AD and BC are parallel. 2◦. Assume now that AB = CD and AB is parallel to CD. Triangles ABC, CDA are again congruent, because they have an equal angle (A1 = C1 are alternate interior angles) between two pairs of equal sides: AC is a common side and AB = CD. From this congruence we again find that A2 = C2, and that sides AD, BC are parallel. 46b. Remark I. A quadrilateral can have two sides AB, CD equal, and the other two BC, AD parallel, without being a parallelogram (it is then called an isosceles trapezoid). Choosing an arbitrary side AB, it suffices to take the line symmetric to AB (D1C, Fig. 45) with respect to any line xy in the plane, provided that the line xy is not parallel to AB, and that xy intersects the line AB in a point I on the extension of AB (and not on segment AB itself). Then quadrilateral ABCD1 will have two parallel sides (both perpendicular to xy) and the other two equal (because they are symmetric to each other). These last two sides are not parallel because they intersect at point I. Conversely, every quadrilateral with two sides BC, AD parallel, and the other two equal is either a parallelogram, or (in the case of an isosceles trapezoid) a figure with a line of symmetry. Indeed, let xy be the perpendicular bisector of BC. We have already found two oblique lines from C to the line AD, both equal to AB, namely: CD1 which is symmetric to AB with respect to xy, and CD2 which,
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