Question

consider the two triangles.
to prove that the triangles are similar by the sas similarity theorem, it needs to be shown that
○ $\angle c \cong \angle c$
○ $\angle c \cong \angle g$
○ $\frac{ac}{gi} = \frac{hi}{bc}$
○ $\frac{ac}{gi} = \frac{bc}{hi}$

Explanation:

Step1: Recall SAS Similarity Rule

SAS similarity requires: 1) A pair of included congruent angles; 2) The ratios of the two pairs of sides forming the angles are equal.

Step2: Identify Corresponding Angles

Both triangles have right angles: $\angle C = 90^\circ$, $\angle I = 90^\circ$, so $\angle C \cong \angle I$. (Note: Option $\angle C \cong \angle G$ is incorrect, $\angle C \cong \angle C$ is irrelevant.)

Step3: Calculate Side Ratios

For $\triangle ABC$ and $\triangle HGI$, the sides forming the right angles are $AC=15, BC=20$ and $HI=12, GI=9$.
Check the ratio of corresponding sides:
$\frac{AC}{GI} = \frac{15}{9} = \frac{5}{3}$
$\frac{BC}{HI} = \frac{20}{12} = \frac{5}{3}$
Thus, $\frac{AC}{GI} = \frac{BC}{HI}$.

Step4: Match to Correct Option

This ratio matches the fourth option, and the included right angles are congruent, satisfying SAS similarity.

Answer:

$\boldsymbol{\frac{AC}{GI} = \frac{BC}{HI}}$