State whether the statement is true or false.
In a rectangle ABCD, if diagonal AC bisects ∠ A as well as ∠ C, then ABCD is a square.

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True

False

answer is A.

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Detailed Solution

Given that, in a rectangle ABCD, if diagonal AC bisects ∠ A as well as ∠ C, then ABCD is a square.
Drawing the figure from the given data.
Question Image

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.
We are given that AC bisects angles A and C.

∠ D C O = ∠ B C O ∠ D A O = ∠ B A O

[1]
The diagonals of a rectangle bisect each other. So, we get:
AO = OC and BO = OD [2]
From the figure, we also get the alternate interior angles,

∠ A C B = ∠ C A D ∠ D C A = ∠ B A C

[3]
From [1] and [3] we get that:

∠ D C O = ∠ B C O = ∠ D A O = ∠ B A O

[4]
In

Δ B A C

∠ B C A = ∠ B A C

. [from 4]
AB = BC [5] [angles opposite to equal angles]
In

Δ D A C

∠ D C A = ∠ D A C

. [from 4]
AD = CD [6] [angles opposite to equal angles]
In

Δ A O B and Δ C O D

,
DO = OB [from 2]
AO = OC [from 2]

∠ A O B = ∠ C O D

[vertically opposite angles]
By SAS congruence criteria,

Δ A O B ≅ Δ C O D

.
We know that the corresponding parts of congruent triangles are equal.
Thus, by C.P.C.T., we get AB = CD [7]
From [5] [6] and [7] we get AB = BC = CD = DA.
A rectangle with all its sides equal is a square. Thus, ABCD is a square. Hence, the given statement is true.
Therefore, the correct option is 1.

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