The diagonal of a dash are equal and bisect each other at 90 degree

1.	The diagonal of a dash are equal and bisect each other at 90 degree
Answer» Square The diagonals of a square are equal and bisect each other at right angles.Detailed proofLet ABCD be a square. Let the diagonals AC and BD intersect each other at a point O. To provethat the diagonals of a square are equal and bisect each other at right angles, we have toprove AC = BD, OA = OC, OB = OD, and ∠AOB = 90º.In ΔABC and ΔDCB,AB = DC (Sides of a square are equal to each other)∠ABC = ∠DCB (All interior angles are of 90)BC = CB (Common side)So, ΔABC ≅ ΔDCB (By SAS congruency)Hence, AC = DB (By CPCT)Hence, the diagonals of a square are equal in length.In ΔAOB and ΔCOD,∠AOB = ∠COD (Vertically opposite angles)∠ABO = ∠CDO (Alternate interior angles)AB = CD (Sides of a square are always equal)So, ΔAOB ≅ ΔCOD (By AAS congruence rule)Hence, AO = CO and OB = OD (By CPCT)Hence, the diagonals of a square bisect each other.In ΔAOB and ΔCOB,As we had proved that diagonals bisect each other, therefore,AO = COAB = CB (Sides of a square are equal)BO = BO (Common)So, ΔAOB ≅ ΔCOB (By SSS congruency)Hence, ∠AOB = ∠COB (By CPCT)However, ∠AOB + ∠COB = 1800 (Linear pair)2∠AOB = 1800∠AOB = 900Hence, the diagonals of a square bisect each other at right angles.

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