Question

Given a parallelogram ABCD. Complete each statement along with the definition used:
$(i)$ $AD=$ $(ii)$ $\mathrm{\angle }DCB=$ $(iii)$ $OC=$ $(iv)$ $m\mathrm{\angle }DAB+m\mathrm{\angle }CDA=$
                                             

Answer 1

Hint: Use the properties of parallelogram: Opposite sides are parallel and equal, opposite angles are equal and sum of adjacent angles is ${180}^{\circ }$.

Complete step-by-step answer:
Given, ACBCD is a parallelogram with O as the point of intersection of its diagonals AC and BD.
$(i)$ We know that in parallelogram, lengths of opposite sides are equal. Hence, we can conclude that:
$\Rightarrow AD=BC$
$(ii)$ We know that in parallelogram, measure of opposite angles is equal. Hence, we can conclude that:
$\Rightarrow \mathrm{\angle }DCB=\mathrm{\angle }DAB$
$(iii)$ We know that in parallelogram, diagonals bisect each other. Hence, we can conclude that:
$\Rightarrow OC=OA$
$(iv)$ We know that in parallelogram, adjacent angles are supplementary (i.e. their sum is ${180}^{\circ }$). Hence, we can conclude that:
$\Rightarrow m\mathrm{\angle }DAB+m\mathrm{\angle }CDA={180}^{\circ }$

Note: In parallelogram, although the diagonals bisect each other but their lengths are not equal.
In the above parallelogram, $AC\ne BD$.