Review of Quadrilaterals

Read this chapter, which summarizes all properties of various quadrilaterals, including the properties of their diagonals.

Rhombus

A rhombus is a quadrilateral with four congruent sides. All rhombuses are parallelograms.

 


The Properties of a Rhombus:

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

2. All four sides are congruent, i.e., \begin{align*}AB = BC = CD = 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. Diagonals are the interior angle bisectors, i.e., \begin{align*}\angle BAC = \angle DAC\end{align*}, \begin{align*}\angle BDC = \angle BDA\end{align*}, \begin{align*}\angle{DBC} = \angle{DBA}\end{align*} and \begin{align*}\angle BCA = \angle DCA\end{align*}.

5. Diagonals intersect each other at right angles, i.e., \begin{align*}\angle AOB\end{align*}\begin{align*}=\end{align*}\begin{align*}\angle BOC\end{align*}\begin{align*}=\end{align*}\begin{align*}\angle COD\end{align*}\begin{align*}=\end{align*}\begin{align*}\angle DOA\end{align*}\begin{align*}= 90^\circ\end{align*}.