Question

the ratios of corresponding sides in the two triangles are equal. what other information is needed to prove that △fge ~ △ijh by the sas similarity theorem? ∠f ≅ ∠j ∠i ≅ ∠f ∠e ≅ ∠h ∠g ≅ ∠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.

For \(\triangle FGE\) and \(\triangle IJH\), we already know that \(\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}\) and \(\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}\). The included angles for these sides in \(\triangle FGE\) and \(\triangle IJH\) are \(\angle F\) and \(\angle I\) respectively. So, if \(\angle I\cong\angle F\), then by the SAS similarity theorem, \(\triangle FGE\sim\triangle IJH\).

Answer:

\(\angle I\cong\angle F\) (i.e., the second option \(\angle I\cong\angle F\))