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

Answer 1

Opal's reasoning contains a few issues in the justification and the manipulation of the proportionality.

1. The proportional relationship derived from similar triangles states that if triangles \( \triangle ADF \sim \triangle ABC \), then the ratios of corresponding sides are equal. Hence, we have \[ \frac{DF}{BC} = \frac{AD}{AB} = \frac{AF}{AC}. \]

2. To establish that \( DF = \frac{1}{2} BC \), Opal should specify that the ratio \( \frac{DF}{BC} = \frac{1}{2} \) based on the similarities derived from the triangles. To fill in the blank correctly, it should read: \[ \frac{DF}{BC} = \frac{1}{2}. \]

3. The manipulation should be: If \( \frac{DF}{BC} = \frac{1}{2} \), then multiplying both sides by \( BC \) leads to \( DF = \frac{1}{2} BC \).

4. For the parallel reasoning, Opal correctly states that if angles \( \angle ADF \) and \( \angle ABC \) are congruent, then the lines \( DF \) and \( BC \) are parallel as per the Corresponding Angles Postulate.

Given these points, the correct fill in the blank option is: \[ DF , BC = \frac{1}{2} \]

This captures the necessary proportionality that allows for the calculation of \( DF \) and supports the conclusion that \( DF \parallel BC \).