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.

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