Question

ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that: (i) ABCD is a square (ii) diagonal BD bisects ∠B as well as ∠D.
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Answer 1

Answer:

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Step-by-step explanation:

(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.