Rhombus

A rhombus is a geometric figure that lies in a plane. It is defined as a quadrilateral all of whose sides are congruent. It is a special type of parallelogram, and its properties (aside from those properties of parallelograms) include:

Its diagonals divide the figure into 4 congruent triangles.
Its diagonals are perpendicular bisectors of eachother.
If all of a rhombus' angles are right angles, then the rhombus is a square.

Proofs

Proof that a rhombus is a parallelogram

All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram.

Or, there is always the longer way:

In rhombus $ABCD$ , all 4 sides are congruent (definition of a rhombus).

$AB\cong CD$ , $BC\cong DA$ , and $AC\cong AC$ .

By the SSS Postulate, $\triangle ABC\cong\triangle CDA$ .

Corresponding parts of congruent triangles are congruent, so $\angle BAC\cong BCA$ and $\angle B\cong\angle D$ . The same can be done for the two other angles, so $\angle A\cong\angle C$ .

Convert the congruences into measures to get $m\angle A=m\angle C$ and $m\angle B=m\angle D$ . Adding these two equations yields $m\angle A+m\angle B=m\angle C+m\angle D$ .

The interior angles of a quadrilateral add up to 360 degrees, so $m\angle A+m\angle B+m\angle C+m\angle D=360$ , or $m\angle A+m\angle B=360-m\angle C-m\angle D$ .

Substituting gives $m\angle C+m\angle D=360-m\angle C-m\angle D$ . When simplified, $m\angle C+m\angle D=180$ .

If two lines are cut by a transversal and same-side interior angles add up to 180 degrees, the lines are parallel. This means $AC\|BD$ . The same can be done for the other two sides, and know we know that opposite sides are parallel. Therefore, a rhombus is a parallelogram.