Question

given \\(\overline{gh} \parallel \overline{jk}\\). enter segments in the blanks provided that would result in a true equation.

answer attempt 1 out of 2
\\(\frac{ki}{hi} = \frac{\square}{\square}\\)

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{GH} \parallel \overline{JK}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle IJK \sim \triangle IGH\) (similar triangles).

Step2: Apply Proportionality of Similar Triangles

In similar triangles, the ratios of corresponding sides are equal. For \(\triangle IJK\) and \(\triangle IGH\), the ratio of \(KI\) to \(HI\) (sides of \(\triangle IJK\) and \(\triangle IGH\)) should be equal to the ratio of \(JI\) to \(GI\) (corresponding sides). So \(\frac{KI}{HI}=\frac{JI}{GI}\).

Answer:

\(\frac{KI}{HI}=\frac{JI}{GI}\) (or other valid corresponding side ratios from similar triangles, e.g., \(\frac{JK}{GH}\) but the side - segment form as above is more in line with the blank structure)