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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 900Square 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 Reasons1) ABCD is a square.1) Given2) AD = BC2) Properties of square.3) ∠BAD = ∠ABC3) Each 900 and by properties of square.4) AB = BA4) Reflexive (common side)5) Δ ADB ≅ ΔBCA5) SAS postulate6) AC = BD6) CPCTC7) OB = OD7) As square is a parallelogram so diagonals of parallelogram bisect each other.8) AB = AD8) Properties of square.9) AO = AO9) Reflexive (common side)10) ΔAOB ≅ ΔAOD10) SSS Postulate11)∠AOB = ∠AOD11) CPCTC12)∠AOB + ∠AOD = 18012) These two angles form linear pair and Linear pair angles are supplementary).13) 2∠AOB = 18013) Addition property.14) ∠AOB = 9014) Division property.15) AO ⊥ BD⇒ AC ⊥ BD15) 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.StatementsReasons1) ABCD is a parallelogram1) Given2) AC = BD and AC ⊥ BD2) Given3) AO = AO3) Reflexive4) ∠AOB = ∠AOD4) Each 9005) OB = OD5) Properties of parallelogram.6) ΔAOB ≅ ΔAOD6) SAS Postulate7) AB = AD7) CPCTC8) AB = CD andAD = BC8) Properties of parallelogram.9) AB = BC = CD = AD9) From above10) AB = AB10) Reflexive (common side)11) AD = BC11) Properties of parallelogram.12) AC = BD12) Given13) ΔABD ≅ Δ BAC13) SSS Postulate14) ∠DAB = ∠CBA14) CPCTC15)∠DAB + ∠CBA = 18015) Interior angles on the same side of the transversal.16) 2∠DAB = 18016) Addition property17) ∠DAB = ∠CBA = 9017) Division property
PracticeHere 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 || ADQuadrilateral• 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 TheoremHome Page

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