To critique Opal's reasoning, let's analyze the process and the statements made in the proof.
Opal correctly begins with the fact that \( \triangle ADF \sim \triangle ABC \), which means that their corresponding sides are proportional. The reasoning for stating that \( DF = \frac{1}{2} BC \) is likely based on the proportionality of the sides. However, the specific way they fill in the blank is important.
The sentence to fill in the blank should emphasize the relationship that arises from the similarity of the triangles. The appropriate reasoning should reflect the relationship between the corresponding sides of similar triangles and incorporate the specific ratio they are dealing with.
Among the available options:
- \( \frac{DF}{BC} = \frac{1}{2} \)
- \( AD = DB \)
- \( DF = BC \)
- \( DF = 2 \)
The first option, \( \frac{DF}{BC} = \frac{1}{2} \), is the most suitable choice to fill in the blank. It shows the relationship that leads to the conclusion \( DF = \frac{1}{2} BC \) after multiplying both sides by segment \( BC \).
With that reasoning in mind, the complete proof would read:
PROOF: Because corresponding sides of similar triangles are proportional, \( \frac{DF}{BC} = \frac{1}{2} \). Multiply both sides of the equation by segment \( BC \) to show that \( DF = \frac{1}{2} BC \). Because corresponding angles of similar triangles are congruent, \( \angle ADF \cong \angle ABC \). If corresponding angles are congruent, then lines are parallel, so \( DF \parallel BC \).
Thus, the answer that should fill in the blank is \( \frac{DF}{BC} = \frac{1}{2} \).
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