Question

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

Explanation:

Step1: Analyze △ABC and △DEF

For right triangles, check if the ratios of corresponding legs are equal. In △ABC, legs are 14 and 20. In △DEF, legs are 8 and 10. Ratios: $\frac{14}{8}=\frac{7}{4}$, $\frac{20}{10}=2$. Not equal, so not similar.

Step2: Analyze △DEF and △GHI

△DEF legs: 8, 10. △GHI legs: 12, 15. Ratios: $\frac{8}{12}=\frac{2}{3}$, $\frac{10}{15}=\frac{2}{3}$. Equal, so similar.

Step3: Analyze △GHI and △ABC

△GHI legs:12,15 (ratio $\frac{12}{15}=\frac{4}{5}$). △ABC legs:14,20 (ratio $\frac{14}{20}=\frac{7}{10}$). Not equal. Wait, recheck △GHI and △JKL: △JKL legs:7,10. △GHI legs:12,15. Ratios: $\frac{7}{12}
eq\frac{10}{15}$. Wait, △JKL and △ABC: △JKL legs 7,10; △ABC legs14,20. Ratios: $\frac{7}{14}=\frac{1}{2}$, $\frac{10}{20}=\frac{1}{2}$. So △JKL ~ △ABC. And △DEF ~ △GHI (from step2), △JKL ~ △ABC (since $\frac{7}{14}=\frac{10}{20}=\frac{1}{2}$). Wait, let's recalculate:

△ABC: legs 14, 20 (right triangle). △JKL: legs 7, 10. Ratio of legs: $\frac{7}{14}=\frac{1}{2}$, $\frac{10}{20}=\frac{1}{2}$. So similar.

△DEF: legs 8,10. △GHI: legs 12,15. $\frac{8}{12}=\frac{2}{3}$, $\frac{10}{15}=\frac{2}{3}$. So similar.

△JKL and △ABC: ratio 1/2, so similar.

Wait, initial step1: △ABC (14,20) and △DEF (8,10): 14/8=7/4, 20/10=2. Not equal. So △ABC ~ △JKL (14/7=2, 20/10=2). So:

• △DEF ~ △GHI (ratios 8/12=2/3, 10/15=2/3)
• △GHI ~ △ABC? No, △GHI (12,15) and △ABC (14,20): 12/14=6/7, 15/20=3/4. Not equal. Wait, △JKL (7,10) and △ABC (14,20): 7/14=1/2, 10/20=1/2. So △JKL ~ △ABC.

So correct pairs: △DEF ~ △GHI, △JKL ~ △ABC, and also check △GHI ~ △JKL? No, △GHI (12,15) and △JKL (7,10): 12/7≠15/10. Wait, let's list all:

• △ABC ~ △JKL: legs 14/7=2, 20/10=2. Similar.
• △DEF ~ △GHI: legs 8/12=2/3, 10/15=2/3. Similar.

So the correct options are △DEF ~ △GHI, △JKL ~ △ABC. Wait, let's re-express:

△ABC: legs 14, 20 (right angle at C).

△DEF: legs 8, 10 (right angle at F).

△GHI: legs 12, 15 (right angle at I).

△JKL: legs 7, 10 (right angle at L).

Check △ABC and △JKL: 14/7 = 2, 20/10 = 2. So similar (SAS similarity, right angle included, ratio of legs equal).

△DEF and △GHI: 8/12 = 2/3, 10/15 = 2/3. So similar (SAS similarity).

△DEF and △ABC: 8/14=4/7, 10/20=1/2. Not equal.

△GHI and △ABC: 12/14=6/7, 15/20=3/4. Not equal.

△GHI and △JKL: 12/7≠15/10.

So the similar pairs are △DEF ~ △GHI, △JKL ~ △ABC.

Answer:

$\triangle DEF \sim \triangle GHI$, $\triangle JKL \sim \triangle ABC$ (i.e., the options: $\triangle DEF \sim \triangle GHI$, $\triangle JKL \sim \triangle ABC$)