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
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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