Rhombus and its Theorems

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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 = 90⁰

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 = 90⁰

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 = 56⁰

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

⇒ ∠ODC = ½ x 56

⇒ ∠ODC = 28⁰

∠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 = 62⁰

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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

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