Question

in $\triangle ghi$ and $\triangle jkl$ below, $\angle g$ is congruent to $\angle j$ and $\angle h$ is congruent to $\angle k$. which of the following is a good first step to prove that $\triangle ghi \sim \triangle jkl$? apply a translation and rotation to $\triangle ghi$ that maps $\angle g$ onto $\angle k$. apply a translation to $\triangle ghi$ that maps $h$ onto $j$. apply a dilation centered at $g$ with scale factor $\frac{jk}{gh}$ to $\triangle ghi$. apply a reflection across $\overline{hi}$ to $\triangle ghi$.

Explanation:

1. The first option incorrectly maps $\angle G$ to $\angle K$, but we know $\angle G \cong \angle J$ and $\angle H \cong \angle K$, so this misaligns congruent angles.
2. The second option maps $H$ to $J$, but $\angle H \cong \angle K$, not $\angle J$, so this does not align matching angles properly.
3. The third option uses a dilation with scale factor $\frac{JK}{GH}$. Since $\angle G \cong \angle J$ and $\angle H \cong \angle K$, aligning the corresponding sides around these congruent angles first by scaling $\triangle GHI$ to match the side length ratio of $\triangle JKL$ is a valid first step to set up similarity via AA (Angle-Angle) after aligning the angles.
4. The fourth option reflects $\triangle GHI$ over $\overline{HI}$, which does not help align the congruent angles or set up the similarity proof.

Answer:

Apply a dilation centered at $G$ with scale factor $\frac{JK}{GH}$ to $\triangle GHI$.