Question

which pairs of triangles are similar? check all that apply. △abc ~ △def △def ~ △ghi △ghi ~ △abc △ghi ~ △jkl △jkl ~ △abc

Explanation:

Step1: Check similarity of \(\triangle ABC\) and \(\triangle DEF\)

For \(\triangle ABC\) and \(\triangle DEF\), \(\frac{14}{8}=\frac{7}{4}\), \(\frac{20}{10} = 2\). Since \(\frac{7}{4}
eq2\), they are not similar.

Step2: Check similarity of \(\triangle DEF\) and \(\triangle GHI\)

For \(\triangle DEF\) and \(\triangle GHI\), \(\frac{8}{12}=\frac{2}{3}\), \(\frac{10}{15}=\frac{2}{3}\). But we need to check the right - angle condition. The sides adjacent to the right - angle in \(\triangle DEF\) are \(8\) and \(10\), in \(\triangle GHI\) are \(12\) and \(15\). Also, \(\frac{8}{12}=\frac{10}{15}\). By the Side - Angle - Side (SAS) similarity criterion (right - angle and proportional sides), they are not similar as the ratio of sides adjacent to the right - angle is \(\frac{2}{3}\) but we need to check all pairs. Wait, no: \(\frac{8}{12}=\frac{2}{3}\), \(\frac{10}{15}=\frac{2}{3}\), but the sides are not in the same order. Let's use the Side - Side - Side (SSS) similarity. We need three pairs of sides. But we only have two pairs of sides (adjacent to the right - angle). So, no.

Step3: Check similarity of \(\triangle GHI\) and \(\triangle ABC\)

For \(\triangle GHI\) and \(\triangle ABC\), \(\frac{14}{15}
eq\frac{20}{12}\) (since \(\frac{14}{15}=\frac{56}{60}\), \(\frac{20}{12}=\frac{100}{60}\)), so they are not similar.

Step4: Check similarity of \(\triangle GHI\) and \(\triangle JKL\)

For \(\triangle GHI\) and \(\triangle JKL\), \(\frac{15}{10}=\frac{3}{2}\), \(\frac{12}{7}
eq\frac{3}{2}\), so they are not similar.

Step5: Check similarity of \(\triangle JKL\) and \(\triangle ABC\)

For \(\triangle JKL\) and \(\triangle ABC\), \(\frac{14}{7} = 2\), \(\frac{20}{10}=2\). By the Side - Angle - Side (SAS) similarity criterion (right - angle and proportional sides: \(\frac{AC}{LJ}=\frac{14}{7}=2\), \(\frac{BC}{KJ}=\frac{20}{10} = 2\) and both have a right - angle), \(\triangle JKL\sim\triangle ABC\)

Answer:

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