Short Answer 
The Side Angle Side (SAS) theorem asserts that if corresponding angles are equal and the sides including these angles are proportional, the triangles are similar. By establishing the proportions of the sides as (dfrac{AC}{BC}) and (dfrac{GI}{HI}), the conclusion confirms that the triangles are similar, represented by the relationship (dfrac{AC}{GI} = dfrac{BC}{HI}), which leads to the correct option being D).
Step 1: Understand the Side Angle Side (SAS) Theorem
The Side Angle Side (SAS) theorem states that if one angle of a triangle is congruent to the corresponding angle of another triangle and the sides including these angles are proportional, then the triangles are similar. This means that the relationships between the sides matter when assessing similarity.
Step 2: Apply the Triangle Relationships
By analyzing the triangles in question, you can establish proportions between the sides. Specifically, you can set up the relationship of the sides as follows:
- (dfrac{AC}{BC}) is proportional to (dfrac{GI}{HI}).
Step 3: Conclude with the Correct Option
From the established proportions using the SAS theorem, it follows that the correct relationship is represented by the statement:
- (dfrac{AC}{GI} = dfrac{BC}{HI}).