PARALLELOGRAM, in geometry, a quadrilateral right-lined figure, whose opposite sides are parallel and equal to each other. It is generated by the equa ble motion of a right line always parallel to itself. When it has all its four angles right, and only its opposite sides equal, it is called a rectangle or oblong. When the angles are all right, and the sides equal, it is called a square. If all the sides are equal, and the angles unequal, it is called a rhombus or ioeenge ; and if the sides and angles be unequal, it is call ed a rhomboides.
In every parallelogram, of what kind soever", a diagonal divides it into two equal parts; the angles diagonally oppo site are equal ; the opposite angles of the same side are together equal to two right angles ; and each two sides, together, greater than the diagonal.
Two parallelograms on the same or equal base, and of the same height, or between the same parallels, are equal ; and hence two triangles on the same base, and of the same height, are also equal. Hence, also, every triangle is halt a pa rallelogram, upon the same or an equal base, and of the same altitude, or between the same parallels. Hence, also, a trian gle is equal to a parallelogram, having the same base, and half the altitude, or half the base, and the same altitude.
Parallelograms, therefore, are in a gi ven ratio, compounded of their bases and altitudes. If then the altitudes be equal, they are as the bases, and conversely.
In similar parallelograms and triangles, the altitudes are proportional to the ho mologous sides, and the bases are cut proportionably thereby. Hence similar parallelograms and triangles are in a du plicate ratio of their homologous sides ; as also of their altitudes, and the seg ments of their bases ; they are, therefore, as the squares of the sides, altitudes, and homologous segments of the bases.
In every parallelogram, the sum of the squares of the two diagonals is equal to the sum of the squares of the four sides. For if the parallelogram be rectangular, it follows that the two diagonals arc equal ; and, consequently, the square of a diagonal, or, which comes to the same thing, the square of the hypothenuse of a right angle, is equal to the squares of the sides. See GEOMETRY.