Theorems Dealing with Parallelograms

Theorems Dealing with Parallelograms

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

*Parallelogram I have: - 2 sets of parallel sides - 2 sets of congruent sides - opposite angles congruent - consecutive angles supplementary - diagonals bisect each other - diagonals form 2 congruent triangles

Using this definition, the remaining properties regarding a parallelogram can be "proven" true and become theorems.

When GIVEN a parallelogram, the definition and theorems are stated as ...

	A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
	If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent. (Proof appears further down the page.)
	If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are congruent.
	If a quadrilateral is a parallelogram, the consecutive angles are supplementary.
	If a quadrilateral is a parallelogram, the diagonals bisect each other.
	If a quadrilateral is a parallelogram, the diagonals form two congruent triangles.

When trying to PROVE a parallelogram, the definition and theorems are stated as ... (many of these theorems are converses of the previous theorems)

	A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
	If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.
	If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.
	If one angle is supplementary to both consecutive angles in a quadrilateral, the quadrilateral is a parallelogram.
	If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
	If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram.  (Proof appears further down the page.)
	** Be sure to remember this last method, as it may save you time when working a proof.

Proof of Theorem:  If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent.
(Remember:  when attempting to prove a theorem to be true,
you cannot use the theorem as a reason in your proof.)

STATEMENTS		REASONS
1		1	Given
2	Draw segment from  A to C	2	Two points determine exactly one line.
3		3	A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
4		4	If two parallel lines are cut by a transversal, the alternate interior angles are congruent.
5		5	Reflexive property:  A quantity is congruent to itself.
6		6	ASA:  If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
7		7	CPCTC:  Corresponding parts of congruent triangles are congruent.

Proof of Theorem:  If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel and congruent, the quadrilateral is a parallelogram. 
(Remember:  when attempting to prove a theorem to be true,
 you cannot use the theorem as a reason in your proof.)

STATEMENTS		REASONS
1		1	Given
2	Draw segment from  A to C	2	Two points determine exactly one line.
3		3	If two parallel lines are cut by a transversal, the alternate interior angles are congruent.
4		4	Reflexive property:  A quantity is congruent to itself.
5		5	SAS:  If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
6		6	CPCTC:  Corresponding parts of congruent triangles are congruent.
7		7	If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
8		8	A parallelogram is a quadrilateral with both pairs of opposite sides parallel.