


A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 

Using the definition, all of the parallelogram properties, when stated as theorems, can be "proven" true.
The properties (theorems) will be stated in "if ...then" form. Both the theorem and its converse (where you swap the "if" and "then" expressions) will be examined.
Click in the charts below to see each proof.
The * means proof is directly referenced in Common Core.
While one method of proof will be shown, other methods are also possible.

Use the following, when GIVEN a parallelogram: 

DEFINITION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 

THEOREM: If a quadrilateral is a parallelogram, it has 2 sets of opposite sides congruent. 
* 



THEOREM: If a quadrilateral is a parallelogram, it has 2 sets of opposite angles congruent. 
* 


THEOREM: If a quadrilateral is a parallelogram, it has consecutive angles which are supplementary. 




THEOREM: If a quadrilateral is a parallelogram, it has diagonals which bisect each other. 
* 


THEOREM: If a quadrilateral is a parallelogram, it has diagonals which form 2 congruent triangles. 



Use the following, to PROVE a parallelogram: 

DEFINITION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 

THEOREM: If a quadrilateral has 2 sets of opposite sides congruent, then it is a parallelogram. 
* 



THEOREM: If a quadrilateral has 2 sets of opposite angles congruent, then it is a parallelogram. 
* 


THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram. 




THEOREM: If a quadrilateral has diagonals which bisect each other, then it is a parallelogram. 
* 



THEOREM: If a quadrilateral has one set of opposite sides which are both congruent and parallel, then it is a parallelogram. 

This last method can save time and energy when working a proof! 

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