Rhombus

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A rhombus is a geometric figure that lies in a plane. It is defined as a quadrilateral all of whose sides are congruent. It is a special type of parallelogram, and its properties (aside from those properties of parallelograms) include:


Proofs

Proof that a rhombus is a parallelogram

All sides of a rhombus are congruent, so opposite sides are congruent, which is one of the properties of a parallelogram.

Or, there is always the longer way:

In rhombus $ABCD$, all 4 sides are congruent (definition of a rhombus).

$AB\cong CD$, $BC\cong DA$, and $AC\cong AC$.

By the SSS Postulate, $\triangle ABC\cong\triangle CDA$.

Corresponding parts of congruent triangles are congruent, so $\angle BAC\cong BCA$ and $\angle B\cong\angle D$, which is one of the properties of a parallelogram (opposite angles are congruent).

Proof that the diagonals of a rhombus divide it into 4 congruent triangles

In rhombus $ABCD$, $M$ is the point at which the diagonals intersect.

Since the diagonals of a rhombus are bisectors of eachother, $AM\cong MC$ and $BM\cong MD$.

Also, all sides are congruent.

By the SSS Postulate, the 4 triangles formed by the diagonals of a rhombus are congruent.

Proof that the diagonals of a rhombus are perpendicular

Continuation of above proof:

Corresponding parts of congruent triangles are congruent, so all 4 angles (the ones in the middle) are congruent.

This leads to the fact that they are all equal to $90^{\circ}$ degrees, and the diagonals are perpendiclar to eachother.

Example Problems

Introductory