Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. in the figure below, for the triangle △ abc and triangle △ def, the corresponding sides are proportional in length so $\frac{ab}{de} = \frac{bc}{ef} = \frac{ac}{df}$. which statement best describes the relationship between these two triangles? they are not similar triangles. they are similar triangles. they are neither similar nor congruent triangles. they are congruent but not similar triangles.

Explanation:

To determine the relationship between triangles \( \triangle ABC \) and \( \triangle DEF \), we use the definition of similar triangles. The Side - Side - Side (SSS) similarity criterion states that if the corresponding sides of two triangles are proportional (i.e., \( \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} \)), then the two triangles are similar.

Let's analyze each option:

• Option "They are not similar triangles": This is incorrect because the given condition of proportional corresponding sides satisfies the SSS similarity criterion.
• Option "They are similar triangles": This is correct. The SSS similarity criterion is met as the ratios of corresponding sides are equal.
• Option "They are neither similar nor congruent triangles": This is incorrect. Since the sides are proportional, they are similar. Congruent triangles are a special case of similar triangles where the ratio of sides is 1, but even if the ratio is not 1, they are still similar.
• Option "They are congruent but not similar triangles": This is incorrect. Congruent triangles are similar (with a scale factor of 1), and the statement says "not similar" which is wrong. Also, we only know the sides are proportional, not necessarily equal (so we can't say they are congruent from the given information, but the main error is the "not similar" part).

Answer:

B. They are similar triangles.