You just saw that the diagonals of a rhombus intersect at a right angle. In other words, they're perpendicular. It turns out that a parallelogram is a rhombus if and only if the diagonals are perpendicular.
Rhombus Postulate
A parallelogram is a rhombus if and only if the diagonals are perpendicular.
Question
In order to prove this postulate, what two supporting statements must you prove?
- If a parallelogram is a rhombus, the diagonals are perpendicular.
- If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.
Keep reading to see how to prove this postulate.
Statement 1
Your Turn
Statement 2
Given: Parallelogram ABCD is a rhombus.
Prove: AC is perpendicular to BD.
| By definition of rhombus, AB \(\small\mathsf{\cong}\) BC \(\small\mathsf{\cong}\) CD \(\small\mathsf{\cong}\) AD because all sides of a rhombus are congruent. |
| By definition of isosceles triangle, ∠a1 \(\small\mathsf{\cong}\) ∠c1 because angles opposite two congruent sides are also congruent. |
| Also by definition of isosceles triangles, ∠b2 \(\small\mathsf{\cong}\) ∠d2, ∠a2 \(\small\mathsf{\cong}\) ∠c2, and ∠b1 \(\small\mathsf{\cong}\) ∠d1 because they are all angles opposite two congruent sides. |
| By angle-side-angle (ASA), △AOB \(\small\mathsf{\cong}\) △BOC \(\small\mathsf{\cong}\) △COD \(\small\mathsf{\cong}\) △AOD. |
| By definition of congruent triangles, ∠AOB \(\small\mathsf{\cong}\) ∠BOC \(\small\mathsf{\cong}\) ∠COD \(\small\mathsf{\cong}\) ∠AOD. |
| By observation ∠AOB + ∠BOC + ∠COD + ∠AOD = 360°. |
| Because the angles are congruent and their sum is 360°, ∠AOB = ∠BOC = ∠COD = ∠AOD = 90° |
| By definition of perpendicular, AC is perpendicular to BD. |
It's your turn. Prove that AC is perpendicular to BD in rhombus ABCD. Create your proof and then check your work by clicking the reveal button below.
Given: Parallelogram ABCD is a rhombus.
Prove: AC is perpendicular to BD.
| Statements | Reasons |
| AB \(\small\mathsf{\cong}\) BC \(\small\mathsf{\cong}\) CD \(\small\mathsf{\cong}\) AD | By definition of rhombus, all sides are congruent |
| ∠a1 \(\small\mathsf{\cong}\) ∠c1, ∠b2 \(\small\mathsf{\cong}\) ∠d2, ∠a2 \(\small\mathsf{\cong}\) ∠c2, and ∠b1 \(\small\mathsf{\cong}\) ∠d1 | By definition of isosceles triangle, angles opposite two congruent sides are also congruent |
| △AOB \(\small\mathsf{\cong}\) △BOC \(\small\mathsf{\cong}\) △COD \(\small\mathsf{\cong}\) △AOD | By angle-side-angle (ASA) |
| ∠AOB \(\small\mathsf{\cong}\) ∠BOC \(\small\mathsf{\cong}\) ∠COD \(\small\mathsf{\cong}\) ∠AOD | By definition of congruent triangles |
| ∠AOB + ∠BOC + ∠COD + ∠AOD = 360° | By observation |
| ∠AOB = ∠BOC = ∠COD = ∠AOD = 90° | The angles are congruent and their sum is 360° |
| AC is perpendicular to BD | By definition of perpendicular |
You've proven the first statement. Now it's time to prove the second statement: If the diagonals of a parallelogram are perpendicular, the parallelogram is a rhombus.
Given: In parallelogram ABCD, AC is perpendicular to BD.
Prove: ABCD is a rhombus.
| By definition of parallel lines, ∠a1 \(\small\mathsf{\cong}\) ∠c2, ∠a2 \(\small\mathsf{\cong}\) ∠ c1, ∠b1 \(\small\mathsf{\cong}\) ∠d2, and ∠b2 \(\small\mathsf{\cong}\) ∠d1 because these are alternate interior angles. |
| By definition of parallelogram, AB \(\small\mathsf{\cong}\) CD and BC \(\small\mathsf{\cong}\) AD because opposite sides of a parallelogram are congruent. |
| By ASA, △BOC \(\small\mathsf{\cong}\) △AOD and △AOB \(\small\mathsf{\cong}\) △COD. |
| By definition of congruent triangles, segment AO \(\small\mathsf{\cong}\) segment OC and segment BO \(\small\mathsf{\cong}\) segment OD because corresponding sides of congruent triangles are congruent. |
| By definition of perpendicular, ∠AOB \(\small\mathsf{\cong}\) ∠BOC because both angles are 90°. |
| By side-angle-side (SAS), △AOB \(\small\mathsf{\cong}\) △BOC. |
| By definition of congruent triangles, side AB \(\small\mathsf{\cong}\) side BC because corresponding sides of congruent triangles are congruent. |
| Because consecutive sides are congruent, parallelogram ABCD is a rhombus. |
