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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¯¯¯¯¯¯¯¯ . (1 point) Responses DF=BC upper D upper F equals upper B upper C DFBC=12 Start Fraction upper D upper F over upper B upper C End Fraction equals Start Fraction 1 over 2 End Fraction BCDF=2 Start Fraction upper B upper C over upper D upper F End Fraction equals 2 AD=DB upper A upper D equals upper D upper B Skip to navigation

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To determine the appropriate response to fill in the blank in Opal's reasoning, let's analyze their reasoning step by step.

  1. Opal states that because corresponding sides of similar triangles are proportional, they can derive the equation involving DF and BC. This is correct for similar triangles; they have proportional sides.

  2. The proof continues by suggesting that to get from the proportionality of the sides to the statement DF = 12BC, they need to multiply both sides of the equation by segment BC.

However, it looks like there are some errors in this reasoning because without knowing the specific proportional relationship (for instance, if DF is stated to be half of BC) you cannot conclude that DF equals 12 times a segment BC. If "DF" is half of "BC" in ratio terms, we obtain DF = (1/2)BC.

Now, looking at the options provided for the blank:

  • "DF=BC"
  • "DF = 12"
  • "Start Fraction DF over BC End Fraction = Start Fraction 1 over 2 End Fraction"
  • "DF = 2 Start Fraction BC over DF End Fraction = 2"
  • "AD = DB, A D = D B"

Given that Opal is trying to establish that DF is half of BC based on the similarity of the triangles, the most logical option to complete the statement would be:

"Start Fraction DF over BC End Fraction = Start Fraction 1 over 2 End Fraction."

This indicates that DF is indeed \( \frac{1}{2} BC \) and shows the proportions clearly, allowing for the statement df=12bc to be derived correctly from it.

Therefore, the correct choice to fill in the blank is:

Start Fraction DF over BC End Fraction = Start Fraction 1 over 2 End Fraction.