# Given a parallelogram ABCD. Complete each statement along with the definition used:

$(i)$ $AD = $ $(ii)$ $\angle DCB = $ $(iii)$ $OC = $ $(iv)$ $m\angle DAB + m\angle CDA = $

Last updated date: 21st Mar 2023

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Answer

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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 \angle DCB = \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\angle DAB + m\angle CDA = {180^ \circ }$

Note: In parallelogram, although the diagonals bisect each other but their lengths are not equal.

In the above parallelogram, $AC \ne BD$.

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 \angle DCB = \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\angle DAB + m\angle CDA = {180^ \circ }$

Note: In parallelogram, although the diagonals bisect each other but their lengths are not equal.

In the above parallelogram, $AC \ne BD$.

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