Question

consider the two triangles.
to prove that the triangles are similar by the sas
similarity theorem, it needs to be shown that
∠c≅∠c
∠c≅∠g
\\(\frac{ac}{gi} = \frac{hi}{bc}\\)
\\(\frac{ac}{gi} = \frac{bc}{hi}\\)

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: Identify Corresponding Sides and Included Angle

In triangle \(ABC\) and triangle \(HIG\) (assuming the right angles are \(\angle C\) and \(\angle I\)), the right angles are the included angles. So we need \(\angle C\cong\angle I\) (but wait, looking at the options, the angles are \(\angle C\) and \(\angle G\) or \(\angle C\) and \(\angle C\)? Wait, no, let's check the side lengths. \(AC = 15\), \(BC=20\), \(HI = 12\), \(GI = 9\). Let's check the ratios.

For SAS similarity, we need \(\frac{AC}{GI}=\frac{BC}{HI}\). Let's calculate the ratios: \(\frac{AC}{GI}=\frac{15}{9}=\frac{5}{3}\), \(\frac{BC}{HI}=\frac{20}{12}=\frac{5}{3}\). And the included angle is \(\angle C\) and \(\angle I\) (both right angles, so \(\angle C\cong\angle I\), but in the options, the angle option is \(\angle C\cong\angle G\) which is wrong, or the side ratio option \(\frac{AC}{GI}=\frac{BC}{HI}\) which is the fourth option.

Answer:

\(\frac{AC}{GI}=\frac{BC}{HI}\) (the fourth option, e.g., if the options are labeled as 1. \(\angle C\cong\angle C\); 2. \(\angle C\cong\angle G\); 3. \(\frac{AC}{GI}=\frac{HI}{BC}\); 4. \(\frac{AC}{GI}=\frac{BC}{HI}\), then the answer is 4. \(\frac{AC}{GI}=\frac{BC}{HI}\))