State whether the statement is true or false.
The diagonals of a square are equal and bisect each other at right angles.

Question

State whether the statement is true or false.The diagonals of a square are equal and bisect each other at right angles.

Accepted Answer

Given that, the diagonals of a square are equal and bisect each other at right angles.Drawing the figure from the given data.ASA congruence criteria (Angle-side-Angle): If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.SAS congruence criteria (side-Angle-side): If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.SSS congruence criteria (side-side-side): Under a given correspondence, two triangles are congruent if the three sides of one triangle are equal to the three corresponding sides of the other.From the figure, in ∆ABC and ∆BCD, line BC is the common line.AB = CD   [sides of square are equal]

∠ABC=∠DCB=
    
     90
    ∘

[angles of square are right angles]By SAS congruence criteria ∆ABC ≅ ∆BCD.We know that the corresponding parts of congruent triangles are equal.Thus, by C.P.T.C., we get AC=BD.In ∆AOB and ∆COD,∠OAB =∠OCD [alternate interior angles when AB||CD and AC is transversal]∠OBA =∠ODC [alternate interior angles when AB||CD and BD is transversal]AB = CD  [sides of a square are equal]By ASA (Angle side angle) congruence criteria, ∆AOB ≅ ∆COD.By C.P.C.T., we get AO = OC and BO = OD……. [1]In ∆AOB and ∆AOD,AB = AD  [sides of a square]OA = OA   [common]OB = OD  [from 1]By SSS congruence criteria, ∆AOB ≅ ∆AOD.Thus, by C.P.C.T., we get ∠AOB =∠AOD……. [2]From figure, we also see that linear pair

∠AOB +∠AOD=
    
     180
    ∘

.From [2] we get,

∠AOB +∠AOD=
      180
      ∘

∠AOB +∠AOB=
      180
      ∘

2∠AOB=
      180
      ∘

∠AOB=
      90
      ∘

∠AOB =∠AOD=
      90
      ∘

Thus, the diagonals of the square are equal and bisect each other at right angles. Hence, the statement is true.Therefore, the correct option is 1.

State whether the statement is true or false.
The diagonals of a square are equal and bisect each other at right angles.

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