Square and its Theorems
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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 .
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1) ABCD is a square. | 1) Given |
2) AD = BC | 2) Properties of square. |
3) ∠BAD = ∠ABC | 3) Each 900 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.
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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 900 |
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
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