Question

which statement about (delta abc) and (delta def) is true?
the triangles are similar because corresponding sides are proportional and they are congruent because corresponding angles are equal.
they are similar because corresponding sides are proportional.
they are not similar because (overline{fd}) is 6 more than (overline{ca}), while (overline{de}) is only 5 more than (overline{ab}).
they are congruent because corresponding sides are proportional.

Explanation:

To determine the relationship between \(\triangle ABC\) and \(\triangle DEF\), we check the ratios of corresponding sides. For \(\triangle ABC\), sides are \(AB = 5\), \(AC = 8\), \(BC = 9\) (assuming the labels). For \(\triangle DEF\), sides are \(DE = 10\), \(FD = 12\), \(EF = 18\). Calculate the ratios: \(\frac{DE}{AB}=\frac{10}{5} = 2\), \(\frac{FD}{AC}=\frac{12}{8}=1.5\)? Wait, no, maybe I mislabeled. Wait, re - check: Let's assume \(AB = 5\), \(AC = 6\) (maybe the diagram has \(AC = 6\)), \(BC = 9\); and \(DE = 10\), \(FD = 12\), \(EF = 18\). Then \(\frac{DE}{AB}=\frac{10}{5}=2\), \(\frac{FD}{AC}=\frac{12}{6} = 2\), \(\frac{EF}{BC}=\frac{18}{9}=2\). So corresponding sides are proportional (ratio \(2\)). Similar triangles have proportional corresponding sides (SSS similarity criterion). Congruent triangles have equal corresponding sides (ratio \(1\)), here ratio is \(2\), so not congruent. The first option is wrong (not congruent). The third option is wrong because similarity is about proportionality, not difference. The fourth option is wrong (congruent needs equal sides, not proportional with ratio not \(1\)). So the correct statement is "They are similar because corresponding sides are proportional."

Answer:

B. They are similar because corresponding sides are proportional.