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Question 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.
Let ABCD be a quadrilateral such that AB = BC = CD = DA and ∠DAB = 90°. Join BD. In ∆ADB and ∆CDB, we have AD = CD, AB = CB, and DB is a common side. ∴ ∆ADB and ∆CDB are congruent. ∴ ∠C = ∠A = 90° In ∆DAB, ∠1 = ∠2 (∵ AB = AD) Also, ∠1 + 90° + ∠2 = 180° ⇒ ∠1 + ∠2 = 90° ⇒ ∠1 = 45° and ∠2 = 45° (∵ ∠1 = ∠2) In ∆CDB, ∠3 = ∠4 (∵ CD = CB) Also, ∠3 + 90° + ∠4 = 180° ⇒ ∠3 + ∠4 = 90° ⇒ ∠3 = ∠4 = 45° (∵ ∠3 = ∠4) ∴ ∠ABC = ∠1 + ∠4 = 45° + 45° = 90° and ∠ADC = ∠2 + ∠3 = 45° + 45° = 90°. ∴ Each angle of the quadrilateral ABCD is 90°. ∴ ABCD is a square. Also, by measurement, we find AB = BC = CD = DA and ∠A = ∠B = ∠C = ∠D = 90°.