Rectangle and its Theorems

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

A rectangle is a parallelogram in which each angle is 90⁰

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 = 90⁰

Prove that : ∠A = ∠B = ∠C = ∠D = 90⁰

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 = 44⁰, find ∠OAD.

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 = 22⁰

As ∠A = 90⁰

∠A = ∠1 + ∠3

90 = 22 + ∠3

So, ∠3 = 90 – 22

∠3 = 68

So, ∠OAD = 68⁰

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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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