Question

$$overline{ab} parallel overline{dc}$$ given
$$angle fab cong angle fcd$$ alternate interior angles theorem
$$angle fba cong angle fdc$$ alternate interior angles theorem
? ?
$$\frac{fb}{fd} = \frac{af}{cf}$$ corresponding sides in similar triangles are proportional
$$\frac{fb}{af} = \frac{fd}{cf}$$ division property of equality
slope of $$overline{ab} = \frac{fb}{af}$$
slope of $$overline{dc} = \frac{fd}{cf}$$ definition of slope
slope of $$overline{ab} =$$ slope of $$overline{dc}$$ substitution property of equality
which step is missing?
a. statement: $$\triangle fdc \sim \triangle fab$$ reason: aa
b. statement: $$\triangle fdc \cong \triangle fba$$ reason: sas
c. statement: $$\triangle fdc \sim \triangle fba$$ reason: aa
d. statement: $$\triangle fdc \cong \triangle fab$$ reason: sas

Explanation:

To determine the missing step, we analyze the given information and the subsequent steps. We know that two pairs of angles are congruent (∠FAB ≅ ∠FCD and ∠FBA ≅ ∠FDC) from the alternate interior angles theorem. For triangle similarity, the AA (Angle - Angle) criterion states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

• Option A: The statement is $\triangle FDC \sim \triangle FAB$ with reason AA. But the angles we have are ∠FAB ≅ ∠FCD and ∠FBA ≅ ∠FDC. So the similar triangles should be $\triangle FDC \sim \triangle FBA$ (since ∠FDC corresponds to ∠FBA and ∠FCD corresponds to ∠FAB), so A is incorrect.
• Option B: The reason given is SAS, but we only have angle information, not side - angle - side information. Also, the triangles are similar, not congruent, so B is incorrect.
• Option C: The statement is $\triangle FDC \sim \triangle FBA$ with reason AA. We have ∠FDC ≅ ∠FBA and ∠FCD ≅ ∠FAB, which satisfies the AA similarity criterion for $\triangle FDC$ and $\triangle FBA$. This makes sense because after establishing the similarity, we can use the proportionality of corresponding sides, which is the next step in the given proof.
• Option D: The reason is SAS, but we don't have the side - angle - side information, and the triangles are similar, not congruent, so D is incorrect.

Answer:

C. Statement: $\triangle FDC \sim \triangle FBA$ Reason: AA