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In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
Consecutive interior angles are the two pairs of angles that: [4] [2] have distinct vertex points, lie on the same side of the transversal and; are both interior. Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°).
The corresponding angles as well as the corresponding sides are defined as appearing in the same sequence, so for example if in a polygon with the side sequence abcde and another with the corresponding side sequence vwxyz we have vertex angle a appearing between sides a and b then its corresponding vertex angle v must appear between sides v and w.
a parallelogram in which a diagonal bisects an interior angle; a parallelogram in which at least two consecutive sides are equal in length; a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram) a quadrilateral with four sides of equal length (by definition)
Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the -norm if and only if =, the so-called Euclidean norm or standard norm. [1] [2]
The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal. The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides). The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas ...
In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to = . Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, as long as θ is not 90°: [18]
The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translation of this parallelogram, [20] a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2 π.