To critique Opal's reasoning and determine which option best fits in the blank, let's clarify the connections made in their proof.
- Understanding Similar Triangles: Since triangles \( \triangle ADF \sim \triangle ABC \), we know that the ratios of their corresponding sides are proportional. Thus, we can write that:
\[ \frac{AD}{AB} = \frac{DF}{BC} = \frac{AF}{AC} \]
- Finding the Correct Relationship: If Opal is trying to prove that \( DF = \frac{1}{2} BC \), and they mention corresponding sides, then we would need to express the relationship between \( DF \) and \( BC \) correctly in the blank.
The ratio given should be comparable and demonstrate that \( DF \) is half of \( BC \). The statement that would correctly fill in the blank is:
"DF/BC = 1/2". (This represents that \( DF \) is half of \( BC \).)
Thus, the correct response from the provided options is:
DFBC=12
Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction
This statement accurately reflects the proportional relationship that leads to the conclusion \( DF = \frac{1}{2} BC \).
Regarding the parallel lines conclusion, since corresponding angles \( \angle ADF \) and \( \angle ABC \) are congruent (as stated), the corresponding sides being in a specific ratio leads to \( DF \parallel BC \) by the converse of the Alternate Interior Angles Theorem.
Opal's reasoning effectively concludes with both the length proportionality and the parallelism correctly stated after filling the appropriate relationship in the blank.