Question

Use the image to answer the question.

A triangle is marked clockwise from the lower left vertex as upper A upper B upper C. Point upper D is marked at the midpoint of side upper A upper B. Point upper F is marked at the midpoint of side upper A upper C.

Monty has already proven that △ADF∼△ABC
. He is now attempting to prove that DF=12BC
and DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯
. Critique his reasoning. Which option should fill in the blank?

Option #1: Because corresponding angles of similar triangles are congruent, ∠ADF≅∠ABC
.

Option #2: Because corresponding angles of similar triangles are congruent, ∠ADF≅∠DAF
.

Option #3: Because corresponding angles are congruent when lines are parallel, ∠ADF≅∠ABC
.

Option #4: Because corresponding angles are congruent when lines are parallel, ∠ADF≅∠DAF
.

(1 point)
PROOF: Because corresponding sides of similar triangles are proportional, DFBC=12
. Multiply both sides of the equation by segment BC
to show that DF=12BC
. Option #
. If corresponding angles are congruent then lines are parallel, so DF¯¯¯¯¯¯¯¯∥BC¯¯¯¯¯¯¯¯
.

Answer 1

Correct choice: Option #1.

Critique: From △ADF ∼ △ABC we get corresponding angles congruent, so ∠ADF ≅ ∠ABC (Option #1). Then by the converse of the corresponding-angles theorem, DF ∥ BC. Also note the side result should read DF = (1/2)·BC (not "12" BC). Options #3 and #4 give the wrong reason (they state the converse direction), and #2 states a false angle correspondence.