You know that a rhombus is a parallelogram with four congruent sides, but how can you prove that a parallelogram is a rhombus? In other words, what is the minimum information you need to know about a parallelogram in order to prove that it's a rhombus? It turns out you only need to know that two consecutive sides of the parallelogram are congruent. Specifically, if a parallelogram has two consecutive congruent sides, then the parallelogram is a rhombus.
Rhombus Postulate
If a parallelogram has two consecutive congruent sides, then the parallelogram is a rhombus.
Prove this postulate using parallelogram ABCD. Click on each step in the table to create your proof.
| Step 1 | Given: ABCD is a parallelogram with AB \(\small\mathsf{\cong}\) BC. |
| Step 2 | Prove ABCD is a rhombus. |
| Step 3 | By definition of parallelogram, opposites sides are congruent. Therefore, AB \(\small\mathsf{\cong}\) CD. |
| Step 4 | Also, by definition of parallelogram, BC \(\small\mathsf{\cong}\) AD. |
| Step 5 | By the transitive property, AB \(\small\mathsf{\cong}\) BC \(\small\mathsf{\cong}\) CD \(\small\mathsf{\cong}\) AD. |
| Step 6 | Because all four sides are congruent, parallelogram ABCD is a rhombus. |
Question
Based on this parallelogram postulate, what's the minimum information you need about the sides of a parallelogram to determine that it's a rhombus?
You need to know that two consecutive sides are congruent.
