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