Question

what additional information is needed to prove that the triangles are similar?
to prove △xyz ~ △mno using sss, you need to know that.
now that you know the length of xz, you need to know that angle o is congruent to to prove △xyz ~ △mno using sas.

Explanation:

Step1: Recall SSS similarity criterion

For SSS (Side - Side - Side) similarity, the ratios of the corresponding sides of two triangles must be equal.
We have \(\frac{XY}{MN}=\frac{21}{5}\), \(\frac{YZ}{NO}=\frac{14}{4}=\frac{7}{2}\). Let \(XZ = x\). For \(\triangle XYZ\sim\triangle MNO\) by SSS, we need \(\frac{XZ}{MO}=\frac{XY}{MN}=\frac{YZ}{NO}\). Since \(MO\) is a side of \(\triangle MNO\), we need to know the length of \(MO\) so that \(\frac{21}{5}=\frac{14}{4}=\frac{XZ}{MO}\)

Step2: Recall SAS similarity criterion

For SAS (Side - Angle - Side) similarity, the ratio of two pairs of corresponding sides must be equal and the included angles must be congruent.
We know the ratio of sides (from the SSS consideration part). The included angle for the sides \(YZ\) and \(XZ\) in \(\triangle XYZ\) is \(\angle Z\), and the included angle for the sides \(NO\) and \(MO\) in \(\triangle MNO\) is \(\angle O\). So if \(\angle O\cong\angle Z\), and the ratios of the corresponding sides \(\frac{YZ}{NO}=\frac{XZ}{MO}\), then \(\triangle XYZ\sim\triangle MNO\) by SAS

Answer:

The length of \(MO\); \(\angle Z\)