Final answer:
Proving the similarity of triangles AXYZ and AMNO using SSS requires the proportions of corresponding sides to be shown. For SAS, after establishing that side XZ is known, proving similarity also requires angle O to be congruent to angle Y.
Step-by-step explanation:
To prove that two triangles are similar using the Side-Side-Side (SSS) criterion, one must show that the corresponding sides of the triangles are in proportion. If you have a triangle AXYZ and another triangle AMNO, you would need to establish that the sides AX/AM, XY/MN, and ZY/NO are all in proportion to conclude that the triangles are similar.
For the Side-Angle-Side (SAS) similarity theorem, you need to know two pairs of corresponding sides are in proportion and the angle between these two sides in one triangle is congruent to the angle between the corresponding sides of the other triangle. In this case, knowing the length of side XZ, you would next need to show that angle O is congruent to angle Y (assuming that Y is the angle between sides AX and XZ in triangle AXYZ) to prove that the triangles AXYZ and AMNO are similar.