Question

the missing statement is \\(\frac{ij}{fh} = \frac{1}{2}\\), and the missing reason is \corresponding sides of similar triangles are proportional.\
the missing statement is \\(\frac{ij}{fh} = 2\\), and the missing reason is \cpctc theorem.\
the missing statement is \\(\frac{ij}{fh} = 2\\), and the missing reason is \corresponding sides of similar triangles are proportional.\
the missing statement is \\(\angle jif \cong \angle gfh\\), and the missing reason is \cpctc theorem.\

Explanation:

1. First, recall the properties of similar triangles: Corresponding sides of similar triangles are proportional. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) applies to congruent, not similar, triangles, so we can eliminate options with CPCTC as a reason.
2. For the ratio: If a triangle is a dilation (scale factor) of another, the ratio of corresponding sides depends on the scale. If \( \triangle IJF \) and \( \triangle GHF \) are similar (assuming context where \( IJ \) is twice \( FH \) or vice - versa), the ratio \( \frac{IJ}{FH} = 2 \) would make sense with the "corresponding sides of similar triangles are proportional" reason. The first option has a ratio of \( \frac{1}{2} \) which would be the reciprocal, and the angle - related option is incorrect as CPCTC is for congruent triangles.

Answer:

The missing statement is \(\boldsymbol{\frac{IJ}{FH}=\frac{2}{1} = 2}\), and the missing reason is "corresponding sides of similar triangles are proportional." (The option: The missing statement is \(\frac{IJ}{FH}=2\), and the missing reason is "corresponding sides of similar triangles are proportional.")