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.
Answers
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.
GIVEN : ABCD is a rectangle in which diagonal Ac bisects ∠ A as well as ∠ C. in rectangle ABCD AD=BC, AB=CD and ∠ A =∠ B = ∠ C= ∠ D = 90°
TO PROVE : (1) ABCD is a square.
(2) Diagonal BD bisects ∠ B as well as ∠ D
PROOF : (1) AB = BC and AB = CD
∠ 1 = ∠ 2 and ∠ 3 = ∠ 4 –––––——— equation 1
∠ 1 = ∠ 4 and ∠ 2 = ∠ 4 ( alternate interior angles ) —————————equation 2
From equation 1 and 2 we get
∠ 1 = ∠ 2 = ∠ 3 = ∠ 4
In ∆ ABC
∠ 2 = ∠ 4
so, AB= BC
In ∆ ACD
∠ 1 = ∠ 3
so, AD = CD
SO, AB=BC=CD=AD and ABCD is a square because all sides of rectangle are equal and hence it is a square.
(2) In ∆ ABD
AB=AD
So, ∠ 5=∠ 8 ----------------- equation3
In ∆ BCD
CD=BC
So, ∠ 6= ∠ 7------------------ equation 4
∠ 5=∠ 7 and ∠ 6 = ∠ 8 ( Alternate interior angles )
from equation 3,4 and 5 we get
∠ 5 = ∠ 6 = ∠ 7 = ∠ 8
so, ∠ 5=∠ 6
and ∠ 7 = ∠ 8
therefore, Diagonal BD bisects ∠ B as well as ∠D
Hence verified.