# Rhombus and its Theorems

In this section we will discuss rhombus and its theorems.Rhombus is a parallelogram with all sides equal and parallel.

Rhombus and its Theorems :

**Theorem 1 : The diagonals of a rhombus are perpendicular to each other.**

**Given :**A rhombus ABCD whose diagonals are AC and BD intersect at O.

**Prove that :**∠BOC = ∠DOC = ∠AOD = ∠AOB = 90

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Statements |
Reasons |

1) ABCD is a rhombus. | 1) Given |

2) AB = BC = CD = DA | 2) Properties of rhombus. |

3) OB = OD and OA = OC | 3) As Parallelogram is a rhombus so diagonal bisect each other. |

4) BO = OD | 4) From (3) |

5) BC = DC | 5) Properties of rhombus. |

6) OC = OC | 6) Reflexive (common side) |

7) ΔBOC ≅ ΔDOC | 7) SSS Postulate. |

8) ∠BOC = ≅ ∠DOC | 8) CPCTC |

9)∠BOC + ∠DOC = 180 | 9) Linear pair angles are supplementary. |

10) 2∠BOC = 180 | 10) Addition property |

11) ∠BOC = 90 | 11) Division property |

12) ∠BOC = ∠DOC = 90 | 12) As these two angles are congruent. |

Hence, ∠AOB = ∠BOC = ∠COD = ∠DOA = 90

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**Converse of the above theorem is also true ⇒ If the diagonals of a parallelogram are perpendicular, then it is a rhombus.**

**Example :**

1) ABCD is a rhombus with ∠ABC = 56. Determine ∠ACD.

**Solution :**ABCD is a rhombus.

∠ ABC = ∠ADC ( Opposite angles are equal)

∠ADC = 56

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∴ ∠ ODC = ½ ∠ADC ( Diagonals of rhombus bisects the angle)

⇒ ∠ODC = ½ x 56

⇒ ∠ODC = 28

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∠OCD + ∠ODC + ∠COD = 180 ( In ΔOCD, sum of all the angles in a triangle is 180)

∠OCD + 28 + 90 = 180

⇒ ∠OCD + 118 = 180

⇒ ∠OCD = 180 -118

∠OCD = 62

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**Quadrilateral**

• Introduction to Quadrilateral

• Types of Quadrilateral

• Properties of Quadrilateral

• Parallelogram and its Theorems

• Rectangle and its Theorems

• Square and its Theorems

• Rhombus and its Theorems

• Trapezoid (Trapezium)and its Theorems

• Kite and its Theorems

• Mid Point Theorem

• Introduction to Quadrilateral

• Types of Quadrilateral

• Properties of Quadrilateral

• Parallelogram and its Theorems

• Rectangle and its Theorems

• Square and its Theorems

• Rhombus and its Theorems

• Trapezoid (Trapezium)and its Theorems

• Kite and its Theorems

• Mid Point Theorem