Three statements are given below: I.In a Rectangle ABCD, the diag… - Vidyadip

MCQ

Three statements are given below:
$I$.In a Rectangle $\text{ABCD}$, the diagonals $AC$ bisects $\angle\text{A}$ as well as $\angle\text{C}.$
$II$.In a Square $\text{ABCD}$, the diagonals $AC$ bisects $\angle\text{A}$ as well as $\angle\text{C}.$
$III$.In rhombus $\text{ABCD}$, the diagonals $AC$ bisects $\angle\text{A}$ as well as $\angle\text{C}.$
Which is True?

✓
$II$ and $III$
B
Only $II$
C
$I$
D
Only $III$

✓

Answer

Correct option: A.

$II$ and $III$

In square and rhombus, all sides are equal.
By joining the points $A$ and $C$ diagonal $AC$ formed.
we get two triangles $A B C$ and $A D C$ which are congruent $(\text{SAS}$ congruence$).$
Also, opposite sides are parallel.
So, by using alternate angle property we can prove that $\angle BAC =\angle DCA$ and $\angle DAC =\angle ACB$.
But by $\text{CPCT} \angle DAC =\angle BAC$.
So, all four angles made by diagonal $A C$ with end points $A$ and $C$ are equal which proves that diagonal $A C$ bisects $\angle A$ and $\angle C$ both.

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