To critique the reasoning in Opal's proof, we need to address the logic behind the statement that corresponds to the blank in the proof.
Given that △ADF∼△ABC, we know the following based on the properties of similar triangles:
- Corresponding sides of similar triangles are proportional.
- Corresponding angles of similar triangles are congruent.
In the context of the question, the blank in the proof is meant to indicate how the proportions of the sides relate to each other.
Critique:
Opal states that "Because corresponding sides of similar triangles are proportional," and the blank should directly express the proportionality that allows us to derive the relationship DF = 1/2 BC.
In this case, the similarity gives us the proportional relationship:
\[ \frac{DF}{BC} = \frac{1}{2} \]
This can be rewritten as:
\[ DF = \frac{1}{2} BC \]
So, the appropriate fill for the blank would be the equation representing this proportionality.
Correct Fill for the Blank:
The best response that fits logically into the blank is:
DF/BC = 1/2
This denotes the relationship of the sides based on the similarity of the triangles. Thus, when considering their reasoning and the requirements for the proof, "DF/BC = 1/2" correctly supports their argument for both DF = 1/2 BC and the implication that DF ∥ BC due to the corresponding angles being congruent.
Updated Proof Structure:
PROOF: Because corresponding sides of similar triangles are proportional, DF/BC = 1/2. Multiply both sides of the equation by segment BC to show that DF = 1/2 BC. Because corresponding angles of similar triangles are congruent, ∠ADF ≅ ∠ABC. If corresponding angles are congruent, then lines are parallel, so DF || BC.
This way, you reinforce both the proportional relationships and the conclusion about the lines being parallel.
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