Question

the ratios of corresponding sides in the two triangles are equal. what other information is needed to prove that \\( \triangle fge \sim \triangle ijh \\) by the sas similarity theorem? \\( \bigcirc \angle f \cong \angle j \\) \\( \bigcirc \angle i \cong \angle f \\) \\( \bigcirc \angle e \cong \angle h \\) \\( \bigcirc \angle g \cong \angle i \\)

Explanation:

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.
In \(\triangle FGE\) and \(\triangle IJH\), we have \(\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}\) and \(\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}\).
For the SAS similarity theorem, we need the included angles of the proportional sides to be congruent. The included angle for sides \(FG\) and \(FE\) in \(\triangle FGE\) is \(\angle F\), and the included angle for sides \(IJ\) and \(IH\) in \(\triangle IJH\) is \(\angle J\). But we need to check the correspondence of the triangles.
Since we want to prove \(\triangle FGE\sim\triangle IJH\), the sides \(FG\) corresponds to \(IJ\) and \(FE\) corresponds to \(IH\). The included angle for the proportional sides should be \(\angle F\) (in \(\triangle FGE\)) and \(\angle I\) (in \(\triangle IJH\)) because of the order of the triangle vertices (\(\triangle FGE\) and \(\triangle IJH\)).

Answer:

\(\angle I\cong\angle F\)