Question

If in rectangle ABCD diagonal AC bisects angle A as well as angle C then ABCD is a

Answer 1

If in rectangle ABCD ,diagonal AC bisects angle A as well as angle C then ABCD is a Square.

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Answer 2

Therefore, $ABCD$ is a square.

Step-by-step explanation:

Given,

Rectangle $ABCD$ where

$AC$ bisects $\angle A$ and $\angle C$

∴$\angle1=\angle2$ and $\angle3=\angle4$

Now,

$AD\parallel BC$ and $AC$ transversal

∴$\angle1=\angle4$  (Alternate angle)

Now,

$\angle1=\angle2$ and $\angle1=\angle4$

Then,

$\angle2=\angle4$

Now,

$\bigtriangleup ABC$

$\angle2=\angle4$

So, $BC=AB$__1  (Side opposite to equal angle are equal)

But, $BC=AD$ and $AB=DC$__2 (Opposite sides of rectangle are equal)

From 1 and 2,

∴$AB=BC=CD=DA$

So, $ABCD$ is a rectangle with all sides equal.

Hence, $ABCD$ is a square.