Answer:
(b) Lines EI and HG are parallel
Step-by-step explanation:
You want to know the statements that are true only if triangles EFI and GFH are similar.
<h3>Similar triangles</h3>
Triangles are similar if corresponding angles are congruent, and corresponding line segments are proportional. So, if triangles EFI and GFH are similar, it means ...
∠E ≅ ∠G ⇒ EI and GH are parallel by alternate interior angles theorem
EI/FI = GH/FH ⇒ corresponding sides are proportional
<h3>True statements</h3>
2FI = 3FH — there is no necessary relation between FI and FH regardless of the similarity of the triangles (false)
EI ║ HG — if these lines are parallel, the triangles must be similar by the alternate interior angles theorem and AA similarity. (true)
E, F, G are collinear — always true, not dependent on similar triangles
EI/FI = GH/FH — given the vertical angles at F are always congruent, this proportion may be true even though other corresponding angles are not congruent. Consider the attached figure, which has EI/FI = 2/3 and both G1H/FH = 2/3 and G2H/FH = 2/3. ∆EFI ~ ∆G1FI, but is not similar to ∆G2FI. (false)
Statement (b) EI║HG is true only if the triangles are similar.
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<em>Additional comment</em>
The proportion EI/FI = GH/FH being true would be equivalent to SSA similarity. In general, this is not a reason for similarity.
However, if the ratio EI/FI > 1, then the proportion would be true only if the triangles were similar. (The attached figure shows the cases possible where EI/FI < 1.)