Question

In each exercise, you could prove that quadrilateral $\mathbf{SANG}$is a parallelogram if one more fact, in addition to those stated were given. State the fact.

$\overline{\mathrm{SZ}}\cong \overline{\mathrm{NZ}}$

Short answer

The additional fact that must be given so that $SANG$can be a parallelogram is that $\mathbf{GZ}$and $\mathbf{AZ}$must be congruent.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that $\overline{SZ}\cong \overline{NZ}$.

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, $\overline{SZ}\cong \overline{NZ}$, 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, $\overline{GZ}\cong \overline{AZ}$.

That implies GZ and AZ must be congruent.

As , $\overline{SZ}\cong \overline{NZ}$, therefore GZ and AZ must be congruent so that $SANG$can be a parallelogram.

Step 3. Description of step.

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