Question

which triangles are similar to \\( \triangle a b c \\)?
choose 1 answer:
\\( \triangle d e f \\) only
\\( \triangle g h i \\) only
both
neither

Explanation:

Step1: Calculate the third angle of $\triangle ABC$

The sum of angles in a triangle is $180^{\circ}$. For $\triangle ABC$, let the third angle be $x$. Then $x = 180^{\circ}-(46^{\circ}+79^{\circ})=180^{\circ}-125^{\circ} = 55^{\circ}$.

Step2: Calculate the third angle of $\triangle DEF$

For $\triangle DEF$, let the third angle be $y$. Then $y=180^{\circ}-(51^{\circ}+62^{\circ})=180^{\circ}-113^{\circ}=67^{\circ}$. Since the angles of $\triangle ABC$ ($46^{\circ},55^{\circ},79^{\circ}$) and $\triangle DEF$ ($51^{\circ},62^{\circ},67^{\circ}$) are not equal, $\triangle DEF$ is not similar to $\triangle ABC$.

Step3: Check the angles of $\triangle GHI$

$\triangle GHI$ has angles $46^{\circ}$ and $79^{\circ}$. Let the third angle be $z$. Then $z = 180^{\circ}-(46^{\circ}+79^{\circ})=55^{\circ}$, which are the same as the angles of $\triangle ABC$. By the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), $\triangle GHI$ is similar to $\triangle ABC$.

Answer:

B. $\triangle GHI$ only