Question

proving similar triangles practice
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marty has already proven that \\( \triangle a d f \sim \triangle a b c \\). he is now attempting to prove that \\( d f=\frac{1}{2} b c \\) and \\( \overline{d f} \\| \overline{b c} \\). critique his reasoning. which option should fill in the blank?
option \\#1 because corresponding angles of similar triangles are congruent. \\( \angle a d f \cong \angle a b c \\)
option \\#2 because corresponding angles of similar triangles are congruent. \\( \angle a d f \cong \angle d a f \\)
option \\#3 because corresponding angles are congruent when lines are parallel. \\( \angle a d f \cong \angle a b c \\)
option \\#4 because corresponding angles are congruent when lines are parallel. \\( \angle a d f \cong \angle d a f \\)
(1 point)

Explanation:

Step1: Analyze the property of similar triangles

For similar triangles, corresponding angles are congruent. In $\triangle ADF\sim\triangle ABC$, $\angle ADF$ and $\angle ABC$ are corresponding angles.

Step2: Check each option

• Option #1: Since $\triangle ADF\sim\triangle ABC$, by the property of similar triangles (corresponding angles of similar triangles are congruent), $\angle ADF\cong\angle ABC$. This is correct.
• Option #2: $\angle ADF$ and $\angle DAF$ are not corresponding angles of similar triangles $\triangle ADF$ and $\triangle ABC$.
• Option #3: The reasoning here is wrong. We are using the property of similar triangles (not parallel - line - angle property) to get the angle congruence at this stage (since we first need to establish the relationship based on similarity).
• Option #4: $\angle ADF$ and $\angle DAF$ are not corresponding angles (either from similar triangles or parallel lines) in a valid way for this proof.

Answer:

Option #1