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
0

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
0 , 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 = 22
0

As ∠A = 90
0

∠A = ∠1 + ∠3

90 = 22 + ∠3

So, ∠3 = 90 – 22

∠3 = 68

So, ∠OAD = 68
0
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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