Theorems About Parallelograms
Rule

Rhombus Diagonals Theorem

A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Parallelogram with diagonals drawn

Based on the diagram, the following relation holds true.

Parallelogram is a rhombus

Proof

This proof will be written in two parts.

If a Parallelogram Is a Rhombus, Then Its Diagonals Are Perpendicular

A rhombus is a parallelogram with four congruent sides. By the Parallelogram Diagonals Theorem, it can be said that its diagonals bisect each other. Let Let be a rhombus with at the midpoint of both diagonals.

Rhombus with diagonals and midpoint that bisect diagonals
Note that is congruent to and is congruent to Additionally, by the Reflexive Property of Congruence, is congruent to itself. Therefore, by the Side-Side-Side Congruence Theorem, is congruent to
Because corresponding parts of congruent triangles are congruent, and are congruent angles. Furthermore, these angles form a linear pair, which means they are supplementary. With this information, it can be concluded that both and are right angles.
This implies that is perpendicular to Therefore, the diagonals of a rhombus are perpendicular.

Parallelogram is a rhombus

If Its Diagonals Are Perpendicular, Then a Parallelogram is a Rhombus

Conversely, let be a parallelogram whose diagonals are perpendicular.

Parallelogram with perpendicular diagonals

By the Parallelogram Diagonals Theorem, the diagonals of the parallelogram bisect each other. If is the midpoint of both diagonals, then and are congruent.

Parallelogram with perpendicular diagonals and mindpoint of diagonals
Since and are perpendicular, and measure and thus are congruent angles. By the Reflexive Property of Congruence, is congruent to itself. This means that two sides and their included angle are congruent. By the Side-Angle-Side Congruence Theorem, and are congruent triangles.
Because corresponding parts of congruent figures are congruent, it can be said that is congruent to
Parallelogram with perpendicular diagonals and mindpoint of diagonals

Furthermore, by the Parallelogram Opposite Sides Theorem, is congruent to and is congruent to By the Transitive Property of Congruence, it follows that all sides of the parallelogram are congruent.

Parallelogram with perpendicular diagonals and mindpoint of diagonals

This means that if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

parallelogram is a rhombus

Exercises