Given two triangles $ABC$ and $DEF$ where $\angle A=\angle D$, $\angle B=\angle E$, and $\frac{AB}{DE}=\frac{BC}{EF}$, which triangles in the diagram are congruent?

• A. No triangles in the diagram are congruent
• B. $ABC$ and $DEF$ are congruent
• C. $ABC$ and $DEF$ are similar but not congruent
• D. $ABC$ and $DFE$ are congruent

Answer: (B)

Verified step by step guidance

1. Identify the given information: angles \( \angle A = \angle D \) and \( \angle B = \angle E \), and the ratio of sides \( \frac{AB}{DE} = \frac{BC}{EF} \).
2. Recall that two triangles are congruent if all corresponding sides and angles are equal, or if they satisfy congruence criteria such as SAS, ASA, or SSS.
3. Note that the given information includes two pairs of equal angles and a proportionality (ratio) between two pairs of sides, not equality of sides.
4. Recognize that having two equal angles and proportional sides corresponds to the criteria for similarity (AA or SAS similarity), not congruence, because side lengths are proportional, not necessarily equal.
5. Conclude that triangles \( ABC \) and \( DEF \) are similar but not necessarily congruent based on the given angle equalities and side ratios.