## 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.

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