Question

ABCD is a rectangle in which diagonal AC bisects angle A as well as angle C. show that ABCD is a square and diagonal BD bisect angle B and angle D​

Answer 1

Answer:

(i)ABCD is a rectangle , in which diagonal AC bisect ∠A as well as ∠C. Therefore,

∠DAC=∠CAB→(1)

∠DCA=∠BCA→(2)

A square is a rectangle when all sides are equal. Now,

AD∥BC & AC is transversal, therefore

∠DAC=∠BCA [Alternate angles]

From (1), ∠CAB=∠BCA→(3)

In △ABC,

∠CAB=∠BCA , therefore

BC=AB →(4)[sides opposite to equal angles]

But BC=AD & AB=DC→(5) [Opposite sides of rectangle]

Therefore from (4)& (5),

AB=BC=CD=AD

Hence, ABCD is a square.

(ii) ABCD is a square and we know that diagonals of a square bisect its

angles.

Hence, BD bisects ∠B as well as ∠D.

Answer 2

Answer:

Step-by-step explanation:

PROOF:-1)Since AC bisects angle A as well as angle C in the rectangle ABCD . Thus, the rectangle ABCD is a square . 2)In a square ,diagonals bisect the angles . So, BD bisects angle B as well as angle D