Question

what additional information is needed to prove that the triangles are similar? to prove \\( \triangle xyz - \triangle 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 \\( \triangle xyz - \triangle mno \\) using sas

Explanation:

Step1: Recall the SSS similarity criterion

For two triangles $\triangle XYZ$ and $\triangle MNO$ to be similar by SSS (Side - Side - Side) similarity, the ratios of their corresponding sides must be equal. That is, if $\triangle XYZ\sim\triangle MNO$ by SSS, then $\frac{XY}{MN}=\frac{YZ}{NO}=\frac{XZ}{MO}$. We know $XY = 21$, $YZ=14$, $NO = 4$. So we need to know the ratio of the third pair of sides in the same proportion as the other two pairs.

Step2: Recall the SAS similarity criterion

For two triangles $\triangle XYZ$ and $\triangle MNO$ to be similar by SAS (Side - Angle - Side) similarity, we need two pairs of sides in proportion and the included angles congruent. We know the lengths of $YZ = 14$, $NO = 4$ and assume the ratio of sides (from the previous SSS - related part). If we use the sides $YZ$ and $NO$ and $XZ$ and $MO$ (after knowing $XZ$), the included angles for these sides in $\triangle XYZ$ and $\triangle MNO$ are $\angle Z$ and $\angle O$ respectively. So if $\angle O\cong\angle Z$, and the sides around these angles are in proportion ($\frac{YZ}{NO}=\frac{XZ}{MO}$), then $\triangle XYZ\sim\triangle MNO$ by SAS.

Answer:

1. $\frac{XY}{MN}=\frac{YZ}{NO}=\frac{XZ}{MO}$
2. $\angle Z$