Criteria proving a quadrilateral is parallelogram
How to Prove that a Quadrilateral Is a Parallelogram
- 1) If a quadrilateral has one pair of sides that are both parallel and congruent.
- 2) If all opposite sides of the quadrilateral are congruent.
- 3) Both pairs of opposite sides are parallel.
- 4) Opposite angles are congruent.
- 5) Diagonals bisect.
If both pairs of opposite sides of a quadrilateral are parallel, then it’s a parallelogram (reverse of the definition).
If both pairs of opposite sides of a quadrilateral are congruent, then it’s a parallelogram (converse of a property).
Tip: To get a feel for why this proof method works, take two toothpicks and two pens or pencils of the same length and put them all together tip-to-tip; create a closed figure, with the toothpicks opposite each other. The only shape you can make is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then it’s a parallelogram (converse of a property).
If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram (converse of a property).
Tip: Take, say, a pencil and a toothpick (or two pens or pencils of different lengths) and make them cross each other at their midpoints. No matter how you change the angle they make, their tips form a parallelogram.
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it’s a parallelogram (neither the reverse of the definition nor the converse of a property).
Tip: Take two pens or pencils of the same length, holding one in each hand. If you keep them parallel, no matter how you move them around, you can see that their four ends form a parallelogram.
A quadrilateral is a parallelogram if:
- Both pairs of opposite sides are parallel. (By definition). Or:
- Both pairs of opposite sides are congruent. If they are congruent, they must also be parallel. Or:
- One pair of opposite sides are congruent and parallel. Then, the other pair must also be parallel.
