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.
PLEASE SOLVE THIS QUESTION OF CLASS 9.
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(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.
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