Parallelograms

Definition

A Parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

• $AB \parallel DC$
• $AD \parallel BC$

Rectangles and squares are special types of parallelograms.

Properties

1. Angles

• Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$.
• Adjacent angles sum to $180^\circ$: Since opposite sides are parallel, consecutive angles are interior angles on the same side of the transversal. $\angle A + \angle B = 180^\circ$

2. Sides

• Opposite sides are equal: $AB = DC$ and $AD = BC$.
  • Proof: Draw a diagonal. It forms two congruent triangles (ASA criterion using alternate interior angles).

3. Diagonals

• Bisect each other: The diagonals intersect at their midpoints. Note that unlike rectangles, the diagonals of a general parallelogram are not equal in length.