If any one of the following conditions is satisfied, the quadrilateral is a parallelogram.
Five Conditions for Being a Parallelogram
- Both pairs of opposite sides are parallel, respectively
- Both pairs of opposite sides are equal, respectively
- Both pairs of opposite angles are equal, respectively
- The diagonals intersect at their respective midpoints
- One pair of opposite sides is parallel and equal
Example ①
In quadrilateral ABCD, when AO=CO and BO=DO (O is the intersection of the diagonals), prove that ABCD is a parallelogram.
Solution: In △AOB and △COD / AO=CO and BO=DO (given) / ∠AOB=∠COD (vertically opposite angles) / Therefore, △AOB≅△COD by SAS. AB=CD and ∠OAB=∠OCD (converse of the alternate interior angle theorem → AB∥CD). Therefore, ABCD is a parallelogram by condition ⑤.
In quadrilateral ABCD, when AO=CO and BO=DO (O is the intersection of the diagonals), prove that ABCD is a parallelogram.
Solution: In △AOB and △COD / AO=CO and BO=DO (given) / ∠AOB=∠COD (vertically opposite angles) / Therefore, △AOB≅△COD by SAS. AB=CD and ∠OAB=∠OCD (converse of the alternate interior angle theorem → AB∥CD). Therefore, ABCD is a parallelogram by condition ⑤.
Key Point
- Showing any one of the five conditions proves that the figure is a parallelogram.
- Condition ④ (the diagonals have the same midpoint) is easy to use in proofs.
Practice Problem
- In quadrilateral ABCD, when AB=DC and AB∥DC, explain using a condition why ABCD is a parallelogram.
Answer and Explanation
- Answer: One pair of opposite sides (AB and DC) is parallel and equal (condition ⑤), so ABCD is a parallelogram.