Question

construct a vertical transversal and horizontal transversal to lines ab and dc that intersect each other at f. label the points of intersection to the parallel lines a, b, c, d.
statements reasons
ab || dc given
∠fab ≅ ∠fcd alternate interior angles theorem
∠fba ≅ ∠fdc alternate interior angles theorem
? ?
fb/fd = af/cf corresponding sides in similar triangles are proportional.
fb/af = fd/cf division property of equality
slope of ab = fb/af definition of slope
slope of dc = fd/cf
slope of ab = slope of dc substitution property of equality
which step is missing?
a. statement: △fdc ~ △fab reason: aa
b. statement: △fdc ~ △fba reason: aa
c. statement: △fdc ≅ △fab reason: sas
d. statement: △fdc ≅ △fba reason: sas

Explanation:

Step1: Analyze the given information

We know $\overline{AB}\parallel\overline{DC}$, $\angle FAB\cong\angle FCD$ and $\angle FBA\cong\angle FDC$ by alternate - interior angles theorem.

Step2: Determine the similarity of triangles

Two pairs of corresponding angles of $\triangle FDC$ and $\triangle FAB$ are congruent. By the AA (angle - angle) similarity criterion, $\triangle FDC\sim\triangle FAB$.

Step3: Check the options

Options C and D are about congruent triangles (using SAS), which is wrong as we are dealing with similar triangles here. Option B has the wrong correspondence of vertices.

Answer:

A. Statement: $\triangle FDC\sim\triangle FAB$ Reason: AA