Question

Opal has already proved that  △ADF∼△ABC . They are now attempting to prove that  DF=12BC  and  DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯ . Critique their reasoning. Which option should fill in the blank? PROOF: Because corresponding sides of similar triangles are proportional, ________________. Multiply both sides of the equation by segment BC to show that  DF=12BC . Because corresponding angles of similar triangles are congruent,  ∠ADF≅∠ABC . If corresponding angles are congruent, then lines are parallel, so  DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯ .

Answer 1

The reasoning provided by Opal has a critical flaw when discussing the proportional relationship between DF and BC.

When transitioning to the proportionality statement, the blank should specify the comparative ratio between the sides of the similar triangles. Since Opal established the similarity of the triangles \( \triangle ADF \sim \triangle ABC \), the lengths of the corresponding sides would be proportional:

CORRECTED PROOF: Because corresponding sides of similar triangles are proportional, \( \frac{DF}{BC} = \frac{AD}{AB} \).

From this statement, you can multiply both sides by \( BC \) to find \( DF \):

CONCLUSION: DF = \frac{AD}{AB} \cdot BC.

If it is given or known that \( AD = \frac{1}{2} AB \), then it follows that \( DF = \frac{1}{2} BC \), supporting their final claims about the lengths.

Finally, the congruency of the angles and therefore the conclusion about the parallel lines is valid. Since \( \angle ADF \cong \angle ABC \) (being corresponding angles from the similar triangles), it is accurate to assert that \( DF \parallel BC \).

Make sure Opal clarifies the ratio based on the corresponding sides clearly in the blank for a complete proof.