Geometry: Parallelograms

On this page we hope to clear up problems that you might have with parallelograms. A parallelogram is a special kind of quadrilateral. There are many special rules and theorems that apply to parallelograms only. By reading on or following any of the links below, you will be on your way to understand parallelograms!

How to tell if a quadrilateral is a paralleogram
Quiz on parallelograms

Theorems that Apply to Parallelograms

A parallelogram is so named because it has two pairs of opposite sides that are parallel.

There are four theorems that apply to parallelograms only. They are outlined below.

1. A diagonal of any parallelogram forms two congruent triangles.

Example:

1.  Problem: Prove triangle ABC in the accompaying figure is congruent 
             to triangle CDA.
   Solution: Since the figure is a parallelogram, segment AB is
             parallel to segment DC and the two segments are also
             congruent.

             Angle 2 is congruent to angle 4 and angle 1 is
             congruent to angle 3.  This is true because alternate
             interior angles are congruent when parallel lines are cut by
             a transversal.

             Segment AC is congruent to segment CA by the
             Reflexive Property of Congruence, which says any figure
             is congruent to itself.

             Triangle ABC is congruent to triangle CDA by
             Angle-Side-Angle.

2. Both pairs of opposite sides of a parallelogram are congruent.

3. Both pairs of opposite angles of a parallelogram are congruent.

4. The diagonals of any parallelogram bisect each other.

Example:

1.  Problem: Prove segment AE in the accompanying figure is congruent to
             segment CE and segment DE is congruent to
             segment BE.
   Solution: By the definition of a parallelogram segment AD and
             segment BC are parallel and congruent.
             
             Angle 1 is congruent to angle 3 and angle 2
             is congruent to angle 4.  This is true because alternate
             interior angles are congruent when parallel lines are cut by
             a transversal.

             Triangle AED and triangle CEB are congruent by
             Angle-Side-Angle.

             The segments we were asked to prove as congruent are
             congruent by CPCTC.

How to Tell if a Quadrilateral is a Parallelogram

In this section we hope to clear up problems associated with figuring out if a given quadrilateral is a parallelogram. Most of the theorems that help us figure out if a shape is a parallelogram are the converses of the theorems stated above.

The three theorems that tell us how to find a parallelogram are outlined below.

1. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

2. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

3. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.

Take the quiz on parallelograms. The quiz is very useful for either review or to see if you've really got the topic down.

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