# How to Prove that a Quadrilateral Is a Rhombus

You can use the following six methods to prove that a quadrilateral is a rhombus. The last three methods in this list require that you first show (or be given) that the quadrilateral in question is a parallelogram:

If all sides of a quadrilateral are congruent, then it’s a rhombus (reverse of the definition).

If the diagonals of a quadrilateral bisect all the angles, then it’s a rhombus (converse of a property).

If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it’s a rhombus (converse of a property).

To visualize this one, take two pens or pencils of different lengths and make them cross each other at right angles and at their midpoints. Their four ends must form a diamond shape — a rhombus.*Tip:*If two consecutive sides of a parallelogram are congruent, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

If either diagonal of a parallelogram bisects two angles, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

If the diagonals of a parallelogram are perpendicular, then it’s a rhombus (neither the reverse of the definition nor the converse of a property).

Here’s a rhombus proof for you. Try to come up with a game plan before reading the two-column proof.

**Statement 1****:**

*Reason for statement 1**:* Given.

**Statement 2****:**

*Reason for statement 2**:* Opposite sides of a rectangle are congruent.

**Statement 3****:**

*Reason for statement 3**:* Given.

**Statement 4****:**

*Reason for statement 4**:* Like Divisions Theorem.

**Statement 5****:**

*Reason for statement 5**:** *All angles of a rectangle are right angles.

**Statement 6****:**

*Reason for statement 6**:** *All right angles are congruent.

**Statement 7****:**

*Reason for statement 7**:** *Given.

**Statement 8****:**

*Reason for statement 8**:** *A midpoint divides a segment into two congruent segments.

**Statement 9****:**

*Reason for statement 9**:** *SAS, or Side-Angle-Side (4, 6, 8)

**Statement 10****:**

*Reason for statement 10**:** *CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

**Statement 11****:**

*Reason for statement 11**:** *Given.

**Statement 12****:**

*Reason for statement 12**:** *If a triangle is isosceles, then its two legs are congruent.

**Statement 13****:**

*Reason for statement 13**:** *Transitivity (10 and 12).

**Statement 14****:**

*Reason for statement 14**:** *If a quadrilateral has four congruent sides, then it’s a rhombus.