Question

question 17 of 50
the triangles are similar by:
diagram of triangles with side lengths 30, 24, 36, 45, 54
options:

• the sas similarity theorem
• the asa similarity theorem
• the sss similarity theorem
• the aa similarity postulate
• none of the choices are correct

Explanation:

Step1: Identify corresponding sides

In triangles \( \triangle ADC \) and \( \triangle ACB \), the sides are:
- \( AD = 30 \), \( AC = 36 \), \( DC = 24 \)
- \( AC = 36 \), \( AB = 45 \), \( CB = 54 \)

Step2: Check ratios of sides

Calculate the ratios of corresponding sides:
- \( \frac{AD}{AC}=\frac{30}{36}=\frac{5}{6} \)
- \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \) Wait, no, correct corresponding sides: \( \frac{AD}{AB}=\frac{30}{45}=\frac{2}{3} \), \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \) Wait, wrong. Correct: \( \triangle ADC \) and \( \triangle ACB \) share \( \angle A \)? Wait, no, check sides around \( \angle A \). Wait, \( AD = 30 \), \( AC = 36 \); \( AC = 36 \), \( AB = 45 \). Wait, \( \frac{AD}{AC}=\frac{30}{36}=\frac{5}{6} \), \( \frac{AC}{AB}=\frac{36}{45}=\frac{4}{5} \)? No, maybe \( \triangle ADC \) and \( \triangle ACB \) with sides \( AD = 30 \), \( DC = 24 \), \( AC = 36 \); and \( AC = 36 \), \( AB = 45 \), \( CB = 54 \). Wait, \( \frac{AD}{AB}=\frac{30}{45}=\frac{2}{3} \), \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \)? No, wait \( \frac{AD}{AC}=\frac{30}{36}=\frac{5}{6} \), \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \)? No, I made a mistake. Wait, correct corresponding sides: \( AD = 30 \), \( AB = 45 \); \( DC = 24 \), \( CB = 54 \); \( AC = 36 \) (common side? No, \( AC \) is a side in both. Wait, \( \frac{AD}{AC}=\frac{30}{36}=\frac{5}{6} \), \( \frac{AC}{AB}=\frac{36}{45}=\frac{4}{5} \)? No, that's not equal. Wait, wait \( \frac{AD}{AB}=\frac{30}{45}=\frac{2}{3} \), \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \)? No, maybe \( \triangle ADC \) and \( \triangle ACB \) with \( AD = 30 \), \( DC = 24 \), \( AC = 36 \); and \( AC = 36 \), \( AB = 45 \), \( CB = 54 \). Wait, \( \frac{AD}{AC}=\frac{30}{36}=\frac{5}{6} \), \( \frac{DC}{CB}=\frac{24}{54}=\frac{4}{9} \)? No, I messed up. Wait, the correct approach: SSS similarity? Wait, calculate ratios:

\( \frac{AD}{AB} = \frac{30}{45} = \frac{2}{3} \)

\( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \) No, wrong. Wait, \( \triangle ADC \) sides: 30, 24, 36; \( \triangle ACB \) sides: 36, 45, 54.

Simplify \( \triangle ADC \) sides: divide by 6: 5, 4, 6. \( \triangle ACB \) sides: divide by 9: 4, 5, 6. Wait, no, 36/9=4, 45/9=5, 54/9=6. And 30/6=5, 24/6=4, 36/6=6. So \( \triangle ADC \) has sides 5,4,6 (scaled by 6) and \( \triangle ACB \) has sides 4,5,6 (scaled by 9)? No, wait 30:45:36? No, 30,24,36 and 36,45,54. Let's find ratios:

\( \frac{30}{45} = \frac{2}{3} \), \( \frac{24}{54} = \frac{4}{9} \), \( \frac{36}{36} = 1 \). No, that's not. Wait, maybe SAS: check two sides and included angle. \( AD = 30 \), \( AC = 36 \); \( AC = 36 \), \( AB = 45 \). So \( \frac{AD}{AC} = \frac{30}{36} = \frac{5}{6} \), \( \frac{AC}{AB} = \frac{36}{45} = \frac{4}{5} \). No, that's not equal. Wait, maybe I got the triangles wrong. The triangles are \( \triangle ADC \) and \( \triangle ACB \), with \( AD = 30 \), \( DC = 24 \), \( AC = 36 \); and \( AC = 36 \), \( CB = 54 \), \( AB = 45 \). Wait, \( \frac{AD}{AB} = \frac{30}{45} = \frac{2}{3} \), \( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \), \( \frac{AC}{AC} = 1 \). No, that's not. Wait, maybe SSS: \( \frac{30}{45} = \frac{2}{3} \), \( \frac{24}{54} = \frac{4}{9} \), \( \frac{36}{36} = 1 \). No. Wait, wait the sides: \( AD = 30 \), \( DC = 24 \), \( AC = 36 \); \( AB = 45 \), \( CB = 54 \), \( AC = 36 \). So \( \frac{AD}{AB} = \frac{30}{45} = \frac{2}{3} \), \( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \), \( \frac{AC}{AC} = 1 \). No, that's not. Wait, maybe I made a mistake. Wait, the correct ratios: \( \frac{AD}{AC} = \frac{30}{36} = \frac{5}{6} \), \( \frac{AC}{AB} = \frac{36}{45} = \frac{4}{5} \). No. Wait, maybe the triangles are \( \triangle ADC \) and \( \triangle BAC \). Wait, no, let's check SSS similarity. For SSS, all three sides must be in proportion.

\( \triangle ADC \) sides: 30, 24, 36.

\( \triangle ACB \) sides: 36, 45, 54.

Divide \( \triangle ADC \) by 6: 5, 4, 6.

Divide \( \triangle ACB \) by 9: 4, 5, 6. Wait, no, 36/9=4, 45/9=5, 54/9=6. And 30/6=5, 24/6=4, 36/6=6. So the sides are 5,4,6 and 4,5,6, which are the same set (just ordered differently). So the ratios are:

\( \frac{30}{45} = \frac{2}{3} \), \( \frac{24}{54} = \frac{4}{9} \)? No, wait 30/54 = 5/9, 24/45=8/15. No, I'm confused. Wait, let's do it properly:

\( \frac{AD}{AB} = \frac{30}{45} = \frac{2}{3} \)

\( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \)

\( \frac{AC}{AC} = 1 \)

No, that's not SSS. Wait, maybe SAS: two sides and included angle. The included angle would be \( \angle A \) for \( \triangle ADC \) (sides AD, AC) and \( \triangle ACB \) (sides AC, AB). So \( \frac{AD}{AC} = \frac{30}{36} = \frac{5}{6} \), \( \frac{AC}{AB} = \frac{36}{45} = \frac{4}{5} \). Not equal. Wait, no, maybe I got the triangles wrong. The triangles are \( \triangle ADC \) and \( \triangle BCA \). Wait, \( AD = 30 \), \( DC = 24 \), \( AC = 36 \); \( BC = 54 \), \( AC = 36 \), \( AB = 45 \). Wait, \( \frac{AD}{BC} = \frac{30}{54} = \frac{5}{9} \), \( \frac{DC}{AC} = \frac{24}{36} = \frac{2}{3} \). No. Wait, maybe the problem is \( \triangle ADC \) and \( \triangle ACB \) with sides \( AD = 30 \), \( AC = 36 \), \( DC = 24 \); and \( AC = 36 \), \( AB = 45 \), \( CB = 54 \). So the sides around \( \angle C \)? No, \( \angle C \) is shared? Wait, no, let's calculate the ratios again:

\( \frac{AD}{AB} = \frac{30}{45} = \frac{2}{3} \)

\( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \)

\( \frac{AC}{AC} = 1 \)

No, this is wrong. Wait, maybe the triangles are similar by SSS. Let's check the ratios of all three sides:

For \( \triangle ADC \) and \( \triangle ACB \):

\( \frac{AD}{AC} = \frac{30}{36} = \frac{5}{6} \)

\( \frac{DC}{CB} = \frac{24}{54} = \frac{4}{9} \)

\( \frac{AC}{AB} = \frac{36}{45} = \frac{4}{5} \)

No, that's not. Wait, maybe I made a mistake in identifying the triangles. The diagram shows \( D \), \( C \), \( B \) with \( DC = 24 \), \( CB = 54 \), \( AD = 30 \), \( AB = 45 \), \( AC = 36 \). So triangles \( \triangle ADC \) and \( \triangle ACB \). Let's list the sides:

\( \triangle ADC \): \( AD = 30 \), \( DC = 24 \), \( AC = 36 \)

\( \triangle ACB \): \( AC = 36 \), \( CB = 54 \), \( AB = 45 \)

Now, sort the sides:

\( \triangle ADC \): 24, 30, 36 (sorted)

\( \triangle ACB \): 36, 45, 54 (sorted)

Now, check ratios:

\( \frac{24}{36} = \frac{2}{3} \)

\( \frac{30}{45} = \frac{2}{3} \)

\( \frac{36}{54} = \frac{2}{3} \)

Ah! There we go. I sorted the sides. So \( \triangle ADC \) (sides 24, 30, 36) and \( \triangle ACB \) (sides 36, 45, 54) when sorted, each ratio is \( \frac{2}{3} \) (24/36=2/3, 30/45=2/3, 36/54=2/3). So all three sides are in proportion, so by SSS similarity theorem.

Step3: Confirm SSS similarity

Since all three corresponding sides are in proportion (\( \frac{24}{36}=\frac{30}{45}=\frac{36}{54}=\frac{2}{3} \)), the triangles are similar by the SSS similarity theorem.

Answer:

the SSS similarity theorem