Conversely, assume the opposite sides of a quadrilateral are equal in length. Draw the diagonal, and the two triangles are congruent by SSS. This means opposite angles are congruent, giving a parallelogram.
A quadrilateral is a polygon with 4 sides, and a parallelogram is a quadrilateral whose opposite sides are parallel. In other words, the top and bottom are parallel and the left and right are parallel.
Label the interior angles w x y z, traveling counterclockwise. Since lines are parallel, w and x are supplementary, and x and y are supplementary, hence w = y. Similarly, x = z. This is an alternate definition of a parallelogram. If catercorner pairs of angles are congruent then adjacent angles sum to 180, and opposite lines are parallel.
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Draw the diagonal from lower left to upper right.
This cuts the parallelogram into two triangles with a common side.
Alternate interior angles are congruent, so we have a side and an angle.
Finally, the corners that do not touch the diagonal are congruent,
hence the triangles are congruent by SAA.
This means opposite sides of a parallelogram are equal in length.
Conversely, assume the opposite sides of a quadrilateral are equal in length. Draw the diagonal, and the two triangles are congruent by SSS. This means opposite angles are congruent, giving a parallelogram. |
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In summary, there are three equivalent characterizations of a parallelogram.
1. Opposite sides are parallel. 2. Opposite angles are congruent. 3. Opposite sides have the same lengths.