Parallelograms and their Properties (Characterization)

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Parallelograms

A parallelogram is a four sided figure (Quadrilateral) with two pairs of opposite sides, each pair being given by parallel line segments of equal length.

Theorem: The sides of a quadrilateral have opposite sides equal of equal length when and only when opposite sides are parallel.

Proof - Part 1: Assume opposite side have equal length. We want to show that opposite sides are parallel.

Draw a diagonal. It divides the quadrilateral into two triangles with a common side (the diagonal). Observe corresponding sides in the triangles are equal. Hence the triangles are isometric by the side-side-side isometry criteria (an earlier assumption or postulate). There corresponding angles in the triangles are equal. The latter implies alternate angles to the diagonal, a transversal for both pairs of opposite sides are equal. The latter equality of alternate equals implies opposite sides are parallel. Therefore opposite sides equal in length. implies opposite sides parallel

Proof - Part 2: Assume opposite side are parallel. We want to show that opposite sides are equal in length.

Draw a diagonal to divide the quadrilateral into two triangles with a common side (the diagonal). The diagonal itself is a transversal for the both the parallel lines through the opposite sides. Hence alternate angles are equal.

The latter implies the triangles are isometric. From the latter, corresponding sides in the triangle have equal lengths. But the corresponding sides in the triangles are opposite sides in the quadrilateral Therefore opposite sides parallel implies opposite sides equal in length.

Theorem: If a pair of opposite sides of a quadrilateral are parallel with equal lengths then the quadrilateral is a parallelogram - that is the other pair of side are parallel to each other and isometric with each other as well.

Proof: Draw a diagonal to divide the quadrilateral

into two triangles with the diagonal as a common side. Alternate angles for the given pair of opposite parallel sides are equal as the diagonal is a transversal between them. Therefore the Side-Angle-Side triangle isometry criteria (assumed earlier) implies the two triangles are isometric.

The quadrilateral is divided into two triangles,
isometric by the side-angle-side criteria.

So corresponding angles and sides in the pair of triangles have equal measure. From which we conclude that the other opposite pair of sides have equal length and that the diagonal is a transversal between them for which the alternate angles are equal. Therefore the other pair of opposite sides are parallel as well. The quadrilateral is thus a parallelogram.