Question

identifying similar triangles
which pairs of triangles are similar? check all that
apply.

Explanation:

Step1: Calculate the ratios of corresponding sides

For \(\triangle ABC\) and \(\triangle DEF\):
\(\frac{AC}{DF}=\frac{14}{8}=\frac{7}{4}\), \(\frac{BC}{EF}=\frac{20}{10} = 2\). Since \(\frac{7}{4}
eq2\), \(\triangle ABC\) and \(\triangle DEF\) are not similar.
For \(\triangle DEF\) and \(\triangle GHI\):
\(\frac{DF}{HI}=\frac{8}{15}\), \(\frac{EF}{GI}=\frac{10}{12}=\frac{5}{6}\). Since \(\frac{8}{15}
eq\frac{5}{6}\), \(\triangle DEF\) and \(\triangle GHI\) are not similar.
For \(\triangle GHI\) and \(\triangle ABC\):
\(\frac{HI}{AC}=\frac{15}{14}\), \(\frac{GI}{BC}=\frac{12}{20}=\frac{3}{5}\). Since \(\frac{15}{14}
eq\frac{3}{5}\), \(\triangle GHI\) and \(\triangle ABC\) are not similar.
For \(\triangle GHI\) and \(\triangle JKL\):
\(\frac{HI}{JL}=\frac{15}{7}\), \(\frac{GI}{KL}=\frac{12}{10}=\frac{6}{5}\). Since \(\frac{15}{7}
eq\frac{6}{5}\), \(\triangle GHI\) and \(\triangle JKL\) are not similar.
For \(\triangle JKL\) and \(\triangle ABC\):
\(\frac{JL}{AC}=\frac{7}{14}=\frac{1}{2}\), \(\frac{KL}{BC}=\frac{10}{20}=\frac{1}{2}\). Also, both have a right - angle. By the Side - Angle - Side (SAS) similarity criterion (ratio of sides and included right - angle), \(\triangle JKL\sim\triangle ABC\).

Answer:

\(\triangle JKL\sim\triangle ABC\)