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
(images of triangles fge and ijh with side lengths: fge has fg=15, fe=30; ijh has ij=10, ih=20)

Explanation:

Step1: Recall SAS Similarity Theorem

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.

Step2: Find Proportional Sides

First, find the ratios of the corresponding sides. For \(\triangle FGE\) and \(\triangle IJH\):

• Side \(FG = 15\), side \(IJ=10\), ratio \(\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}\)
• Side \(FE = 30\), side \(IH = 20\), ratio \(\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}\)

So, the sides \(FG\) and \(FE\) in \(\triangle FGE\) are proportional to sides \(IJ\) and \(IH\) in \(\triangle IJH\) (with ratio \(\frac{3}{2}\)). The included angle between \(FG\) and \(FE\) is \(\angle F\), and the included angle between \(IJ\) and \(IH\) is \(\angle I\)? Wait, no. Wait, let's re - identify the sides. Wait, in \(\triangle FGE\), the sides adjacent to \(\angle F\) are \(FG\) and \(FE\). In \(\triangle IJH\), the sides adjacent to \(\angle I\) are \(IJ\) and \(IH\). Wait, no, let's check the labels again. Wait, the triangles are \(\triangle FGE\) and \(\triangle IJH\). Let's list the vertices:

For \(\triangle FGE\): vertices \(F\), \(G\), \(E\)

For \(\triangle IJH\): vertices \(I\), \(J\), \(H\)

We have \(FG = 15\), \(IJ = 10\); \(FE=30\), \(IH = 20\). So \(\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}\) and \(\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}\). So the two sides of \(\triangle FGE\) ( \(FG\) and \(FE\)) are proportional to two sides of \(\triangle IJH\) ( \(IJ\) and \(IH\)). The included angle for \(\triangle FGE\) between \(FG\) and \(FE\) is \(\angle F\), and the included angle for \(\triangle IJH\) between \(IJ\) and \(IH\) is \(\angle I\)? Wait, no, maybe I mixed up the sides. Wait, maybe the sides are \(FG\) and \(FJ\)? No, the second triangle is \(IJH\). Wait, maybe the correct correspondence is \(FG\) corresponds to \(IJ\), \(GE\) corresponds to \(JH\), and \(FE\) corresponds to \(IH\). Wait, let's re - calculate the ratios.

Wait, \(FG = 15\), \(IJ = 10\), ratio \(\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}\)

\(FE=30\), \(IH = 20\), ratio \(\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}\)

So the sides \(FG\) and \(FE\) of \(\triangle FGE\) are proportional to sides \(IJ\) and \(IH\) of \(\triangle IJH\). The included angle between \(FG\) and \(FE\) is \(\angle F\), and the included angle between \(IJ\) and \(IH\) is \(\angle I\)? Wait, no, in \(\triangle IJH\), the sides adjacent to \(\angle I\) are \(IJ\) and \(IH\). In \(\triangle FGE\), the sides adjacent to \(\angle F\) are \(FG\) and \(FE\). So for SAS similarity, we need the included angles to be congruent. So \(\angle F\) (included angle of \(FG\) and \(FE\)) should be congruent to \(\angle I\) (included angle of \(IJ\) and \(IH\)). So we need \(\angle I\cong\angle F\)

Answer:

\(\boldsymbol{\angle I \cong \angle F}\) (the option: \(\boldsymbol{\angle I \cong \angle F}\))