# 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.

## Contents

## 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 , all 4 sides are congruent (definition of a rhombus).

, , and .

By the SSS Postulate, .

Corresponding parts of congruent triangles are congruent, so and . The same can be done for the two other angles, so .

Convert the congruences into measures to get and . Adding these two equations yields .

The interior angles of a quadrilateral add up to 360 degrees, so , or .

Substituting gives . When simplified, .

If two lines are cut by a transversal and same-side interior angles add up to 180 degrees, the lines are parallel. This means . 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 , is the point at which the diagonals intersect.

Since the diagonals of a rhombus are bisectors of eachother, and .

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.