Question

angle similarity: sss and sas
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?
$\angle i \cong \angle f$
$\angle g \cong \angle i$
$\angle f \cong \angle j$
$\angle e \cong \angle h$

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 angle between those two sides is congruent, then the two triangles are similar.

Step2: Find the ratios of corresponding sides

First, we find the ratios of the given sides. For $\triangle FGE$ and $\triangle IJH$:

• In $\triangle FGE$, $FG = 15$ and $FE=30$. In $\triangle IJH$, $IJ = 10$ and $IH = 20$.
• Calculate the ratios: $\frac{FG}{IJ}=\frac{15}{10}=\frac{3}{2}$ and $\frac{FE}{IH}=\frac{30}{20}=\frac{3}{2}$. So the sides $FG$ and $FE$ of $\triangle FGE$ are proportional to sides $IJ$ and $IH$ of $\triangle IJH$ with a ratio of $\frac{3}{2}$.

Step3: Identify the included angle

The included angle between $FG$ and $FE$ in $\triangle FGE$ is $\angle F$. The included angle between $IJ$ and $IH$ in $\triangle IJH$ is $\angle I$. For the SAS similarity theorem, we need the included angles to be congruent. So we need $\angle F\cong\angle I$ (or $\angle I\cong\angle F$).

Looking at the options:

• Option 1: $\angle I\cong\angle F$ is the angle we need (the included angle between the proportional sides).
• Option 2: $\angle G\cong\angle I$: $\angle G$ is not the included angle between $FG$ and $FE$, so this is incorrect.
• Option 3: $\angle F\cong\angle J$: $\angle J$ is not the included angle between $IJ$ and $IH$, so this is incorrect.
• Option 4: $\angle E\cong\angle H$: $\angle E$ is not the included angle between $FG$ and $FE$, so this is incorrect.

Answer:

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