Question

Question 15
ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.

Answer 1

Given In a rectangle ABCD, diagonal BD bisects ∠B.

Construct Join AC.

To show ABCD is a square.

Proof

In $\Delta BADand\Delta BCD$, $\angle ABD=\angle CBD$ [given]
$\angle A=\angle C$ [each ${90}^{\circ }$]
And $BD=BD$ [common side]

$\therefore$ $\Delta BAD=\Delta BCD$ [by AAS congruence rule]

$\therefore$ AB=BC

and BC=CD [by CPCT rule]..(i)

But in rectangle ABCD, opposite sides are equal.

$\therefore$ AB=CD

and BC­ =AD

From Eqs. (i) and (ii),

AB=BC=CD=DA

So, ABCD is a square. Hence proved.