# 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:

## 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 $AD\|BC$. 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.

### Proof that the diagonals of a rhombus divide it into 4 congruent triangles

In rhombus $ABCD$, $M$ is the point at which the diagonals intersect.

Since the diagonals of a rhombus are bisectors of eachother, $AM\cong MC$ and $BM\cong MD$.

Also, all sides are congruent.

By the SSS Postulate, the 4 triangles formed by the diagonals of a rhombus are congruent.

### Proof that the diagonals of a rhombus are perpendicular

Continuation of above proof:

Corresponding parts of congruent triangles are congruent, so all 4 angles (the ones in the middle) are congruent.

This leads to the fact that they are all equal to 90 degrees, and the diagonals are perpendicular to each other.