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

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.

Step2: Identify sides and angles

In $\triangle ABC$ and $\triangle GIH$, the right angles are included angles. For SAS similarity, we need the ratio of corresponding sides.
$AC = 15$, $BC=20$, $GI = 9$, $HI = 12$.
Check the ratio of sides:
$\frac{AC}{GI}=\frac{15}{9}=\frac{5}{3}$ and $\frac{BC}{HI}=\frac{20}{12}=\frac{5}{3}$.
So, $\frac{AC}{GI}=\frac{BC}{HI}$ satisfies the side - side part of the SAS similarity theorem (with the right angles as the included congruent angles).

Answer:

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