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).
Tip: 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.
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.
![image0.png](https://www.dummies.com/wp-content/uploads/258563.image0.png)
![image1.jpg](https://www.dummies.com/wp-content/uploads/258564.image1.jpg)
Statement 1:
![image2.png](https://www.dummies.com/wp-content/uploads/258565.image2.png)
Reason for statement 1: Given.
Statement 2:
![image3.png](https://www.dummies.com/wp-content/uploads/258566.image3.png)
Reason for statement 2: Opposite sides of a rectangle are congruent.
Statement 3:
![image4.png](https://www.dummies.com/wp-content/uploads/258567.image4.png)
Reason for statement 3: Given.
Statement 4:
![image5.png](https://www.dummies.com/wp-content/uploads/258568.image5.png)
Reason for statement 4: Like Divisions Theorem.
Statement 5:
![image6.png](https://www.dummies.com/wp-content/uploads/258569.image6.png)
Reason for statement 5: All angles of a rectangle are right angles.
Statement 6:
![image7.png](https://www.dummies.com/wp-content/uploads/258570.image7.png)
Reason for statement 6: All right angles are congruent.
Statement 7:
![image8.png](https://www.dummies.com/wp-content/uploads/258571.image8.png)
Reason for statement 7: Given.
Statement 8:
![image9.png](https://www.dummies.com/wp-content/uploads/258572.image9.png)
Reason for statement 8: A midpoint divides a segment into two congruent segments.
Statement 9:
![image10.png](https://www.dummies.com/wp-content/uploads/258573.image10.png)
Reason for statement 9: SAS, or Side-Angle-Side (4, 6, 8)
Statement 10:
![image11.png](https://www.dummies.com/wp-content/uploads/258574.image11.png)
Reason for statement 10: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Statement 11:
![image12.png](https://www.dummies.com/wp-content/uploads/258575.image12.png)
Reason for statement 11: Given.
Statement 12:
![image13.png](https://www.dummies.com/wp-content/uploads/258576.image13.png)
Reason for statement 12: If a triangle is isosceles, then its two legs are congruent.
Statement 13:
![image14.png](https://www.dummies.com/wp-content/uploads/258577.image14.png)
Reason for statement 13: Transitivity (10 and 12).
Statement 14:
![image15.png](https://www.dummies.com/wp-content/uploads/258578.image15.png)
Reason for statement 14: If a quadrilateral has four congruent sides, then it’s a rhombus.