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 the 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.

Step2: Identify the sides and angles

In $\triangle ABC$ and $\triangle HIG$, $\angle C$ and $\angle I$ are right angles ($\angle C=\angle I = 90^{\circ}$).
We have $AC = 15$, $BC=20$, $HI = 12$, $GI = 9$.
We need to check the ratio of the sides.
For the SAS similarity, we should check the ratio of the sides adjacent to the right angles.
The ratio of the sides:
$\frac{AC}{GI}=\frac{15}{9}=\frac{5}{3}$ and $\frac{BC}{HI}=\frac{20}{12}=\frac{5}{3}$
So, to prove $\triangle ABC\sim\triangle HIG$ by SAS similarity theorem, we need to show $\frac{AC}{GI}=\frac{BC}{HI}$

Answer:

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