Question

ABCD is a parallelogram. If two diagonals are equal, what is the measure of angle ABC?

• A. $45°$
• B. $60°$
• C. $90°$
• D. $180°$

Answer 1

The correct option is C $90°$

In parallelogram $ABCD$,
$AD=BC$
$AB=CD$
and $BD=AC$ (given)
In $\Delta ACD$ and $\Delta BDC$
$AD=BC$
$AC=BD$
$CD=CD$
$\therefore \Delta ACD\cong \Delta BDC$ (by SSS congruence criterion)
$\angle ADC=\angle BCD$ (CPCT)
$\therefore AD||BC$
$\angle ADC+\angle BCD={180}^0$ (Alternate-interior angles)
$\Rightarrow 2\angle ADC={180}^0$
We know that the angles opposite to each other in a parallelogram are equal. So, $\angle ADC=\angle ABC=90°$
Thus, the correct answer is option (c).