91Ó°ÊÓ

The given diagram is:

It is being given that $\stackrel{¯}{SZ}â‰\dots \stackrel{¯}{NZ}$.

If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.

From the given diagram it can be observed that the diagonals of the quadrilateral $SANG$are$SN$and $AG$.

As, $\stackrel{¯}{SZ}â‰\dots \stackrel{¯}{NZ}$, therefore it can be noticed that Z is the midpoint of SN.

That implies the diagonal SN is bisected at point Z.

Therefore, the quadrilateral SANG can be a parallelogram if another diagonal AG gets bisected at the point Z.

Therefore, $\stackrel{¯}{GZ}â‰\dots \stackrel{¯}{AZ}$.

That implies GZ and AZ must be congruent.

As , $\stackrel{¯}{SZ}â‰\dots \stackrel{¯}{NZ}$, therefore GZ and AZ must be congruent so that $SANG$can be a parallelogram.

Therefore, the additional fact that must be given so that $SANG$can be a parallelogram is that GZ and AZ must be congruent.