Question 8 - Figure it out - Page 107 - Chapter 4 Class 8 - Quadrilaterals (Ganita Prakash)

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Question 8 If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.Since all four sides are equal It is a rhombus Let’s make diagonal BD and mark angles ∠ 1 to ∠ 4 In ∆ DAB & ∆ DCB DA = DC AB = CB DB = DB ∴ ∆ DAB ≅ ∆ DCB By CPCT (Corresponding Parts of Congruent Triangles) ∠ C = ∠ A = 90° In ∆ DAB Since AD = AB By Isosceles triangle property Angles opposite to equal sides are equal ∴ ∠ 1 = ∠ 3 Also, In ∆ DCB Since CD = CB By Isosceles triangle property Angles opposite to equal sides are equal ∴ ∠ 2 = ∠ 4 Also, in ∆ DAB By Angle sum property ∠1 + ∠A + ∠3 = 180° Putting ∠ 1 = ∠ 3 ∠ 1 + 90° + ∠1 = 180° 2 × ∠1 = 90° ∠1 = (90°)/2 ∠1 = 45° Thus, ∠ 3 = ∠ 1 = 45° Similarly, ∠ 2 = ∠ 4 = 45° So, ∠ D = ∠ 1 + ∠ 2 = 45° + 45° = 90° ∠ B = ∠ 3 + ∠ 4 = 45° + 45° = 90° Thus, ABCD has all 4 sides equal and all 4 angles 90° ∴ It is a square