Question

consider the two triangles shown.
which statement is true?
the given sides and angles cannot be used to show
similarity by either the sss or sas similarity
theorems.
the given sides and angles can be used to show
similarity by the sss similarity theorem only.
the given sides and angles can be used to show
similarity by the sas similarity theorem only.
the given sides and angles can be used to show
similarity by both the sss and sas similarity
theorems.

Explanation:

Step1: Check SAS Similarity

First, identify the included angles (the angles between the sides) are equal (marked as equal in the diagram). Then, find the ratios of the corresponding sides. For triangle \( \triangle JKL \) and \( \triangle FGH \):

• \( \frac{JL}{FH}=\frac{8}{36}=\frac{2}{9} \)? Wait, no, let's pair correctly. Wait, \( JL = 8 \), \( FH = 36 \); \( KL = 9 \), \( FG = 32 \); \( JK = 12 \), \( GH = 48 \). Wait, no, let's check the sides around the equal angle. In \( \triangle JKL \), sides around the angle: \( JL = 8 \), \( KL = 9 \). In \( \triangle FGH \), sides around the equal angle: \( FH = 36 \), \( FG = 32 \)? Wait, no, maybe I mixed up. Wait, the angle at \( L \) and angle at \( F \) are equal? Wait, the diagram: \( \triangle JKL \) has sides \( JK = 12 \), \( KL = 9 \), \( JL = 8 \). \( \triangle FGH \) has \( GH = 48 \), \( FG = 32 \), \( FH = 36 \). Let's find ratios:

For SAS: The two sides around the equal angle should be in proportion. Let's see, angle at \( L \) (in \( \triangle JKL \)) and angle at \( F \) (in \( \triangle FGH \)) are equal (marked as the same angle symbol). So sides around angle \( L \): \( JL = 8 \), \( KL = 9 \). Sides around angle \( F \): \( FH = 36 \), \( FG = 32 \)? Wait, no, maybe \( JL \) corresponds to \( FH \), \( KL \) corresponds to \( FG \)? Wait, \( \frac{JL}{FH}=\frac{8}{36}=\frac{2}{9} \), \( \frac{KL}{FG}=\frac{9}{32} \)? No, that's not equal. Wait, maybe I got the correspondence wrong. Let's check the other sides. \( JK = 12 \), \( GH = 48 \): \( \frac{12}{48}=\frac{1}{4} \). \( KL = 9 \), \( FG = 32 \): \( \frac{9}{32} \)? No. Wait, \( JK = 12 \), \( FH = 36 \): \( \frac{12}{36}=\frac{1}{3} \). \( KL = 9 \), \( FG = 32 \): no. Wait, maybe \( JL = 8 \), \( FG = 32 \): \( \frac{8}{32}=\frac{1}{4} \). \( KL = 9 \), \( FH = 36 \): \( \frac{9}{36}=\frac{1}{4} \). Ah! So the angle at \( L \) (in \( \triangle JKL \)) is between \( JL = 8 \) and \( KL = 9 \), and angle at \( F \) (in \( \triangle FGH \)) is between \( FG = 32 \) and \( FH = 36 \). So the ratios: \( \frac{JL}{FG}=\frac{8}{32}=\frac{1}{4} \), \( \frac{KL}{FH}=\frac{9}{36}=\frac{1}{4} \). So the two sides around the equal angle are in proportion (\( \frac{1}{4} \)) and the included angle is equal, so SAS similarity holds.

Now check SSS: Let's find all three ratios. \( \frac{JK}{GH}=\frac{12}{48}=\frac{1}{4} \), \( \frac{JL}{FG}=\frac{8}{32}=\frac{1}{4} \), \( \frac{KL}{FH}=\frac{9}{36}=\frac{1}{4} \). Wait, \( JL = 8 \), \( FG = 32 \): \( 8/32 = 1/4 \). \( KL = 9 \), \( FH = 36 \): \( 9/36 = 1/4 \). \( JK = 12 \), \( GH = 48 \): \( 12/48 = 1/4 \). So all three sides are in proportion (ratio \( 1/4 \)), so SSS similarity also holds. Wait, so both SAS and SSS? Wait, let's recheck the SSS ratios:

\( JK = 12 \), \( GH = 48 \): \( 12/48 = 1/4 \)

\( JL = 8 \), \( FG = 32 \): \( 8/32 = 1/4 \)

\( KL = 9 \), \( FH = 36 \): \( 9/36 = 1/4 \)

Yes! So all three sides are in proportion (SSS), and the included angle for SAS is equal with sides in proportion (since the two sides around the angle: for \( \triangle JKL \), sides \( JL = 8 \), \( KL = 9 \); for \( \triangle FGH \), sides \( FG = 32 \), \( FH = 36 \). Wait, \( 8/32 = 1/4 \), \( 9/36 = 1/4 \), so the two sides around the equal angle are in proportion (1/4) and the included angle is equal, so SAS also holds. So both SSS and SAS similarity theorems apply.

Step2: Conclusion

So the given sides and angles can be used to show similarity by both the SSS and SAS similarity theorems.

Answer:

The given sides and angles can be used to show similarity by both the SSS and SAS similarity theorems.