In the parallelogram , the line as a transversal cuts the parallel lines and , whence by the theorem of the parent entry (http://planetmath.org/CorrespondingAnglesInTransversalCutting) the alternate interior angles and are congruent. And since the line also cuts the parallel lines and , the alternate interior angles and are congruent. Moreover, the triangles and have a common side . Thus, these triangles are congruent (ASA). Accordingly, the corresponding sides are congruent: and . ∎
Theorem 2. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
Theorem 3. If one pair of opposite sides of a quadrilateral are both parallel and congruent, the quadrilateral is a parallelogram.
Theorem 4. The diagonals of a parallelogram bisect each other.
Theorem 5. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
|Date of creation||2013-03-22 17:15:37|
|Last modified on||2013-03-22 17:15:37|
|Last modified by||pahio (2872)|
|Synonym||properties of parallelograms|