# Rectangle and its Theorems

Rectangle and its Theorems :On the basis of its properties, there are different theorems.

A rectangle is a parallelogram in which each angle is 90
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Rectangle and its Theorems :

Theorem 1 : Each of the four angles of a rectangle is a right angle. Given : A rectangle ABCD, such that ∠A = 90 0

Prove that : ∠A = ∠B = ∠C = ∠D = 90 0
 Statements Reasons 1) ABCD is a rectangle. 1) Given 2)∴ ABCD is a Parallelogram. 2) Every rectangle is a Parallelogram. 3) AD || BC 3) By Properties of parallelogram. 4) ∠A + ∠B = 1800 4) Interior angles on the same side of transversal are supplementary. 5) 90 + ∠B = 180 5) ∠A = 90 (Given) 6) ∠B = 900 6) By subtraction property. 7) ∠D= 90 and ∠C= 90 7) By properties of parallelogram.

Theorem 2 : The diagonals of a rectangle are of equal length. Given : A rectangle ABCD with AC and BD are its diagonals.

Prove that : AC = BD
 Statements Reasons 1) ABCD is a rectangle. 1) Given 2) AD = BC 2) Property of rectangle (opposite sides are equal) 3) AB = AB 3) Reflexive (common side) 4) ∠A = ∠B 4) Each right angle.(property of rectangle) 5) Δ DAB ≅ Δ CBA 5) SAS Postulate 6) AC = BD 6) CPCTC

Example :

1) The diagonals of a rectangle ABCD meet at ‘O’. If ∠BOC = 44

Solution : ∠ BOC + ∠BOA = 180 [ Linear pair angles are supplementary]

⇒ 44 + ∠ BOA = 180

⇒ ∠BOA = 180 – 44

⇒ ∠ BOA = 136

As diagonals of a rectangle are equal and bisect each other.

So, OA = OB

⇒ ∠1 = ∠2

∠1 + ∠2 + ∠BOA = 180

2∠1 + 136 = 180

2∠1 = 180 -136

2∠1 = 44

∴ ∠1 = 22
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As ∠A = 90
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∠A = ∠1 + ∠3

90 = 22 + ∠3

So, ∠3 = 90 – 22

∠3 = 68

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