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 900

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 .

Statements
Reasons
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.
Theorem 2 : If the diagonals of a parallelogram are equal and intersect at right angles, then the parallelogram is a square.


Given : ABCD is parallelogram in which AC = BD and AC ⊥ BD.

Prove that : ABCD is a square.

Statements
Reasons
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

Practice

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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