Opal has already proved that ADF ~ ABC. They are now attempting to prove that DF = 1/2 BC and DF || BC. Critique their reasoning. Which option should fill in the blank?

Question

PROOF: Because corresponding sides of similar triangles are proportional, ____. 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.

BC/DF=2
AD=DB
DF=BC
DF/BC=1/2

Answer 1

To critique Opal's reasoning for proving that \( DF = \frac{1}{2} BC \) and \( DF || BC \), we need to break down their proof step by step.

1. Similarity and Proportional Sides: Opal correctly states that because triangles \( ADF \) and \( ABC \) are similar, the corresponding sides are proportional. Thus, we can write: \[ \frac{DF}{BC} = \frac{AD}{AB} = \frac{AF}{AC} \]

2. Next Steps in the Proof: The attempt to multiply both sides of the equation by segment \( BC \) is correct, but the subsequent statements need clarification:

   • It looks like they are using the proportion incorrectly. Simply stating \( DF = \frac{1}{2} BC \) is not automatically valid without providing the reasoning behind the proportional relationship (i.e., specifying that \( AD \) is half of \( AB \) or another valid reasoning).

3. Congruent Angles and Parallel Lines: They assert that \( \angle ADF \) is congruent to \( \angle ABC \) because the triangles are similar. However, they need to state clearly that if \( \angle ADF \) is congruent to \( \angle ABC\), then line \( DF \) is parallel to line \( BC \) due to the Converse of the Corresponding Angles Postulate. This step requires proper justification.

4. Inconsistencies in Statements: The lines like "BC/DF=2" and "DF=BC" contradict their earlier points and are confusing because they don't align with the context of similar triangles, especially when trying to prove \( DF = \frac{1}{2} BC \).

To fill in the blank logically in the context of the given proof, you could use a statement linking the relationships based on similarity.

A suitable option to fill in would be:

"DF corresponds to half of BC due to the proportions established by similar triangles."

This fills the gap in their proof regarding proportionality, corrects the reasoning related to why \( DF \) is half of \( BC \), and keeps the focus on the properties of similar triangles.

To conclude, ensure that your proof clearly establishes why the sides are proportional through a proper claim, avoids contradictory statements, and emphasizes the justification for parallel lines based on those angles.

Answer 2

use one of the four answers provided

Answer 3

Based on the options provided in your original request, the best choice to fill in the blank is:

"DF/BC = 1/2."

This statement accurately reflects the proportional relationship among the sides of the similar triangles \( ADF \) and \( ABC \). By stating this, they can then justify that multiplying both sides by \( BC \) leads to \( DF = \frac{1}{2} BC \).

So the complete reasoning could read as:

"Because corresponding sides of similar triangles are proportional, DF/BC = 1/2. Multiply both sides of the equation by segment \( BC \) to show that \( DF = \frac{1}{2} BC\)."

This establishes their argument correctly, justifying both the length relationship and supporting their conclusion about parallel lines based on corresponding angles.