Answer :


Consider the parallelogram ABCD with diagonals AC and BD as shown and they intersect at right angles at O


A parallelogram is a square if its adjacent angles are 90° and adjacent sides are equal


So we have to prove that adjacent sides of given parallelogram are equal and adjacent angles are 90° to prove given parallelogram is a square


Consider ΔBAD and ΔCDA


AC = BD … given


AB = DC … opposite sides of a parallelogram


AD is the common side


Therefore, ΔBAD ΔCDA … SSS test for congruency


BAD = CDA …corresponding angles of congruent triangles …(i)


As it is given that ABCD is parallelogram


BAD + CDA = 180° … sum of adjacent angles of a parallelogram is 180°


Using equation (i)


CDA + CDA = 180°


2 × CDA = 180°


CDA = 90°


BAD = 90°


Thus, adjacent angles are 90° … (iii)


Consider ΔAOB and ΔAOD


AOB = AOD … both 90° because given that diagonals intersect at right angles


OD = OB … diagonals of a parallelogram bisect each other


AO is the common side


Therefore, ΔAOB ΔAOD … SAS test for congruency


AB = AD … corresponding sides of congruent triangles


Thus, adjacent sides are equal … (iv)


From statements (iii) and (iv) we can conclude that parallelogram ABCD is a square


Therefore, if in a parallelogram, the diagonals are equal in lengths and intersect at right angles, the parallelogram will be a square.


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