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 = 180^{0} | 4) Interior angles on the same side of transversal are supplementary. |
5) 90 + ∠B = 180 | 5) ∠A = 90 (Given) |
6) ∠B = 90^{0} | 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