Square and its Theorems
In this section we will discuss square and its theorems.A square is a parallelogram with all sides equal and all angles are 90 ^{0}
Square and its Theorems :
Theorem 1 : The diagonals of a square are equal and perpendicular to each other.
Given : ABCD is a square.
Prove that : AC = BD and AC ⊥ BD .


1) ABCD is a square.  1) Given 
2) AD = BC  2) Properties of square. 
3) ∠BAD = ∠ABC  3) Each 90^{0} and by properties of square. 
4) AB = BA  4) Reflexive (common side) 
5) Δ ADB ≅ ΔBCA  5) SAS postulate 
6) AC = BD  6) CPCTC 
7) OB = OD  7) As square is a parallelogram so diagonals of parallelogram bisect each other. 
8) AB = AD  8) Properties of square. 
9) AO = AO  9) Reflexive (common side) 
10) ΔAOB ≅ ΔAOD  10) SSS Postulate 
11) ∠AOB = ∠AOD 
11) CPCTC 
12) ∠AOB + ∠AOD = 180 
12) These two angles form linear pair and Linear pair angles are supplementary). 
13) 2∠AOB = 180  13) Addition property. 
14) ∠AOB = 90  14) Division property. 
15) AO ⊥ BD ⇒ AC ⊥ BD 
15) Definition of perpendicular. 
Given : ABCD is parallelogram in which AC = BD and AC ⊥ BD.
Prove that : ABCD is a square.


1) ABCD is a parallelogram  1) Given 
2) AC = BD and AC ⊥ BD  2) Given 
3) AO = AO  3) Reflexive 
4) ∠AOB = ∠AOD  4) Each 90^{0} 
5) OB = OD  5) Properties of parallelogram. 
6) ΔAOB ≅ ΔAOD  6) SAS Postulate 
7) AB = AD  7) CPCTC 
8) AB = CD and AD = BC 
8) Properties of parallelogram. 
9) AB = BC = CD = AD  9) From above 
10) AB = AB  10) Reflexive (common side) 
11) AD = BC  11) Properties of parallelogram. 
12) AC = BD  12) Given 
13) ΔABD ≅ Δ BAC  13) SSS Postulate 
14) ∠DAB = ∠CBA  14) CPCTC 
15)∠DAB + ∠CBA = 180  15) Interior angles on the same side of the transversal. 
16) 2∠DAB = 180  16) Addition property 
17) ∠DAB = ∠CBA = 90  17) Division property 
Here is a square drawn for you. Answer the following questions on the basis of square and its theorems ( m > measure ).
a. (i) m∠A =  (ii) m∠B =  (iii) m∠C = 
b. (i) seg(AB) =  (ii) seg (BC) =  (iii) seg (CD) = 
C. (i) seg(AC) =  (ii) seg (BD) =  (iii) seg (BO) = 
d. (i) seg(AO) =  (ii) seg (CO) = 
e. (i)m∠DOA =  (ii) m∠AOB =  (iii) m∠BOC = 
f. (i) Is AB  CD (ii) Is BC  AD
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