Final answer:
Two triangles are similar by the SAS criterion if two sides of one are in proportion to two sides of the other and the included angles are congruent. The relationship NO f / A₁B₁ = AB / A₁B₁ must hold true for triangles ABC and DEF to be similar by SAS similarity.
Step-by-step explanation:
For two triangles to be similar by the SAS (Side-Angle-Side) criterion, it means that the two sides of one triangle are in proportion to the two corresponding sides of the other triangle, and the included angle between those sides is congruent in both triangles. When considering the triangles ABC and DEF, we must have that the ratio of the lengths of two sides of triangle ABC to the corresponding two sides of triangle DEF is constant, and the angle included between the two sides in both of the triangles is the same.
In the case of the triangles being represented algebraically as BAO and B₁A₁O, the ratio NO f / A₁B₁ = AB / A₁B₁ needs to be true. This relationship suggests that for triangle ABC to be similar to DEF by SAS similarity, one set of corresponding sides must have equal ratios, and the angle included by those sides must be congruent.