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 .

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

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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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