Theorems Dealing with Parallelograms
Topic Index | Geometry Index | Regents Exam Prep Center

 

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.