If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the are similar (SAS similarity criterion).
Step-by-Step Solution
Key Concept: Use the SAS (Side‑Angle‑Side) similarity criterion: when two sides of one triangle are in the same ratio as the corresponding two sides of another triangle and the included angles are equal, the triangles are similar.
1. Given\
Let \(\triangle ABC\) and \(\triangle DEF\) be two triangles such that\
\[\angle B = \angle E\]\
and the sides surrounding these angles satisfy\
\[\frac{AB}{DE}=\frac{BC}{EF}=k\] (the same constant \(k\)).\
2. Construct a triangle\
On side \(DE\) of \(\triangle DEF\) construct a point \(G\) such that \(DG = k\cdot DE = AB\).\
Join \(G\) to \(E\). Then \(\triangle DGE\) has \(DG = AB\) and \(DE = DE\) with the included angle \(\angle DGE = \angle B\).\
3. Apply the SAS congruence\
Since \(\triangle DGE\) and \(\triangle ABC\) have two sides equal respectively and the included angle equal, they are congruent (SAS). Hence\
\[\angle D = \angle A \quad\text{and}\quad \angle G = \angle C\].\
4. Relate \(\triangle DGE\) to \(\triangle DEF\)\
Because \(DG = AB\) and \(GE = BC\) are in the same ratio as \(DE\) and \(EF\), the triangle \(\triangle DGE\) is a scaled copy of \(\triangle DEF\). Therefore\
\[\frac{DG}{DE}=\frac{GE}{EF}=k\] and \(\angle DGE = \angle B\).\
5. Conclusion\
From steps 3 and 4 we deduce that \(\triangle ABC\) and \(\triangle DEF\) have all corresponding angles equal, i.e.,\
\[\angle A = \angle D,\; \angle B = \angle E,\; \angle C = \angle F.\]\
Hence the two triangles are similar by the definition of similarity.
Result: The given condition (one equal angle and the surrounding sides in the same ratio) guarantees that the two triangles are similar. This is exactly the SAS similarity criterion.
Correct Answer: The two triangles are similar (SAS similarity criterion).