Properties of Rhombus

Key Concepts

• Define a rhombus.
• Explain the conditions required for a parallelogram to be a rhombus.

Rhombus 

A parallelogram with four equal sides is called a Rhombus. 

Theorem 

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. 

Given: WY⊥XZ

To prove: WXYZ is a rhombus. 

Proof: Let us name the parallelogram as WXYZ 

In △WOX and △WOZ

WO=WO [Reflexive property] 

∠WOX=∠WOZ [Right angles] 

OX=OZ [Diagonals of a parallelogram bisect each other] 

So, △WOX≅ △WOZ [by Side-Angle-Side congruence criterion] 

Then, XW=ZW [Congruent parts of congruent triangles] 

We know that the opposite sides of the parallelogram are equal. 

Therefore,

WX = XY = YZ = ZW

Hence Proved.

Theorem 

If a diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. 

Given: ∠1=∠2; ∠3=∠4

To prove: PQRS is a rhombus. 

Proof: The opposite sides of the given figure are parallel to each other. 

So, PQRS is a parallelogram. 

In △PQR and △PSR,

∠1=∠2

PR=PR [Reflexive property] 

∠3=∠4

So, △PQR≅ △PSR [ASA congruence criterion] 

Therefore,

PQ=PS and QR=SR  [Corresponding parts of congruent triangles] 

But in a parallelogram, the opposite sides are equal. 

Therefore,

PQ = QR = RS = SP

So, 𝑷𝑸𝑹𝑺 is a rhombus

Theorem 

If a parallelogram is a rhombus, then the diagonals are perpendicular bisectors of each other. 

Given: ST=TU=UV=VS

To prove: SU⊥TVSU⊥TV

Proof:  In △VKU and △TKU

VK=TK [Diagonals bisect each other] 

KU=KU [Reflexive property] 

UV=TU [Sides of a rhombus are equal] 

So, △VKU ≅ △TKU△VKU ≅ △TKU [SSS congruence criterion] 

If two triangles are congruent, then the corresponding angles are congruent. 

Therefore, ∠VKU=∠TKU

We know that ∠VKU∠VKU and ∠TKU∠TKU form a linear pair.

So, ∠VKU+∠TKU=180°

∠VKU+∠VKU=180°

2 ∠VKU=180°

∠VKU=90°

Hence, ∠TKU=90°

The diagonals are perpendicular to each other.

Theorem 

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. 

Given: ABCD is a rhombus 

To prove: AC bisects ∠DAB and ∠BCD, BD bisects ∠ABC and ∠CDA

Proof: In △AOB and △AOD,

AO=AO [Reflexive property] 

OB=OD [Diagonals bisect each other] 

AB=AD [All sides of a rhombus are equal] 

So, △AOB≅ △AOD [By SSS congruence criterion] 

Now, ∠BAO=∠DAO [Congruent parts of congruent triangles]    

So, AC bisects ∠DAB

Similarly, AC bisects ∠BCD and BD bisects ∠ABC and ∠CDA.

Hence Proved.

Exercise

• Quadrilateral STUV is a rhombus; find the values of x and y.

• If the given quadrilateral is a rhombus, find MNO

• Find the value of x if the given quadrilateral is a rhombus?

• If PQRS is a rhombus, what is the length of QT?

• Find the angle measure of ABC for rhombus ABCD.

Concept Map

What we have learned

• If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
• If a diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.