State whether the statement is true or false.
In a rhombus ABCD, the diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B, as well as ∠D.

Question

State whether the statement is true or false. In a rhombus ABCD, the diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B, as well as ∠D.

Accepted Answer

Given that, in a rhombus ABCD, the diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B, as well as ∠D.The given figure depicts the given situation.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.The sides of a rhombus are equal and the opposite sides and angles of a rhombus are equal and parallel.The diagonals of a rhombus bisect each other at right angles, so we get:OA = OC  [1]OB = OD   [2]

∠AOB=∠BOC=∠COD=∠AOD=
    
     90
    0

[3]In ∆AOD and ∆COD,OA = OC  [from 1]OD = OD  [common]

∠AOD=∠COD
   [from 3]By SAS congruence criteria, we get

ΔAOD≅ΔCOD
  .We know that the corresponding parts of congruent triangles are equal.Thus, by C.P.C.T., we get

∠5=∠6
  .  [4]Similarly,

ΔBOC≅ΔDOC
  .Thus, by C.P.C.T., we get

∠7=∠8
  .  [5]Similarly,

ΔBOC≅ΔBOA
  .Thus, by C.P.C.T., we get

∠1=∠2
  .  [6]Similarly,

ΔAOB≅ΔAOD
  .Thus, by C.P.C.T., we get

∠3=∠4
  .  [7]From the figure, we also get that:

∠B=∠1+∠2

∠A=∠4+∠3

∠D=∠5+∠6

∠C=∠7+∠8

From the relations [4] [5] [6] and [7] we get,

1
      2
     
     ∠B=∠1=∠2

1
      2
     
     ∠A=∠4=∠3

1
      2
     
     ∠D=∠5=∠6

1
      2
     
     ∠C=∠7=∠8

Thus, the diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B, as well as ∠D. Hence, the given statement is true.Therefore, the correct option is 1.

State whether the statement is true or false.
In a rhombus ABCD, the diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B, as well as ∠D.

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