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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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True

False

answer is A.

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Detailed Solution

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.
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.
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]

∠ A O B = ∠ B O C = ∠ C O D = ∠ A O D = 900

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

∠ A O D = ∠ C O D

[from 3]
By SAS congruence criteria, we get

Δ A O D ≅ Δ C O D

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

∠ 5 = ∠ 6

. [4]
Similarly,

Δ B O C ≅ Δ D O C

.
Thus, by C.P.C.T., we get

∠ 7 = ∠ 8

. [5]
Similarly,

Δ B O C ≅ Δ B O A

.
Thus, by C.P.C.T., we get

∠ 1 = ∠ 2

. [6]
Similarly,

Δ A O B ≅ Δ A O D

.
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,

12 ∠ B = ∠ 1 = ∠ 2 12 ∠ A = ∠ 4 = ∠ 3 12 ∠ D = ∠ 5 = ∠ 6 12 ∠ 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.

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