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
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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