Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.


 
The Properties of a Parallelogram:

1. Opposite sides are parallel, i.e., \(\begin{align*}AB \parallel CD\end{align*}\) and \(\begin{align*}BC \parallel DA\end{align*}\).

2. Opposite sides are congruent, i.e., \(\begin{align*}AB = CD\end{align*}\) and \(\begin{align*}BC = DA\end{align*}\).

3. Opposite angles are congruent, i.e., \(\begin{align*}\angle{ABC} = \angle{CDA}\end{align*}\) and \(\begin{align*}\angle{DAB} = \angle{BCD}\end{align*}\).

4. Adjacent angles are supplementary, i.e., \(\begin{align*}\angle{ABC} + \angle{BCD} = 180^\circ\end{align*}\), \(\begin{align*}\angle{BCD} + \angle{CDA} = 180^\circ\end{align*}\), \(\begin{align*}\angle{CDA} + \angle{DAB} = 180^\circ\end{align*}\)and \(\begin{align*}\angle{DAB} + \angle{ABC} = 180^\circ\end{align*}\).

5. Diagonals bisect each other, i.e., \(\begin{align*}BO = OD\end{align*}\) and \(\begin{align*}AO = OC\end{align*}\).

6. Each diagonal bisects the parallelogram, i.e., divides it into two congruent triangles \(\begin{align*}\triangle ABC \cong \triangle ADC\end{align*}\) and \(\begin{align*}\triangle BCD \cong \triangle BAD\end{align*}\).