Parallelogram

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

• Opposite sides are equal in length.
• Opposite angles are equal.
• Consecutive angles are supplementary (sum to $180°$).
• Diagonals bisect each other.

Area: $A = b \cdot h$ where $b$ is any side (base) and $h$ is the perpendicular height — same as a rectangle (parallelogram is a sheared rectangle, and shearing preserves area).

Special parallelograms:

• Rectangle: all angles right.
• Rhombus: all sides equal.
• Square: both rectangle and rhombus.

In vector terms, two vectors $\vec{u}, \vec{v}$ from a common point determine a parallelogram with area $|\vec{u} \times \vec{v}|$ (cross product magnitude). This is the geometric meaning of the determinant.