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). In triangle \( FGH \) and \( LJK \), the angles at \( F \) and \( L \) are equal (marked as equal angles). Now, check the ratios of the sides around these angles. For triangle \( FGH \): sides \( FH = 36 \), \( FG = 32 \); for triangle \( LJK \): sides \( LJ = 8 \), \( LK = 9 \)? Wait, no, let's label correctly. Wait, triangle \( HFG \): \( HF = 36 \), \( FG = 32 \), \( HG = 48 \). Triangle \( KJL \): \( KJ = 12 \), \( JL = 8 \), \( KL = 9 \). Wait, the included angle: in \( \triangle HFG \), angle at \( F \); in \( \triangle KJL \), angle at \( L \). Wait, let's find the ratios of the sides adjacent to the equal angles.

For SAS: the two sides around the equal angle should be in proportion. Let's see: \( \frac{HF}{KJ} = \frac{36}{12} = 3 \), \( \frac{FG}{JL} = \frac{32}{8} = 4 \)? Wait, no, maybe I mixed up the sides. Wait, no, let's check the sides: \( FH = 36 \), \( FG = 32 \); \( LJ = 8 \), \( LK = 9 \)? Wait, no, the triangle \( KJL \): \( KJ = 12 \), \( JL = 8 \), \( KL = 9 \). The triangle \( HFG \): \( HF = 36 \), \( FG = 32 \), \( HG = 48 \). Wait, the equal angles: angle at \( F \) (between \( HF = 36 \) and \( FG = 32 \)) and angle at \( L \) (between \( LJ = 8 \) and \( LK = 9 \))? No, wait, maybe the sides are \( HF = 36 \), \( HG = 48 \), \( FG = 32 \); and \( KJ = 12 \), \( KL = 9 \), \( JL = 8 \). Wait, let's compute the ratios of the sides:

For SSS: check all three sides. \( \frac{HF}{KJ} = \frac{36}{12} = 3 \), \( \frac{FG}{JL} = \frac{32}{8} = 4 \), \( \frac{HG}{KL} = \frac{48}{9} = \frac{16}{3} \)? No, that can't be. Wait, no, I must have mixed up the sides. Wait, maybe the triangle \( HFG \): \( HF = 36 \), \( FG = 32 \), \( HG = 48 \). Triangle \( KJL \): \( KJ = 12 \), \( JL = 8 \), \( KL = 9 \). Wait, no, let's check the ratios again. Wait, \( \frac{HF}{KL} = \frac{36}{9} = 4 \), \( \frac{FG}{JL} = \frac{32}{8} = 4 \), \( \frac{HG}{KJ} = \frac{48}{12} = 4 \). Oh! Wait, I had the sides wrong. So \( HF = 36 \), \( KL = 9 \): \( 36/9 = 4 \); \( FG = 32 \), \( JL = 8 \): \( 32/8 = 4 \); \( HG = 48 \), \( KJ = 12 \): \( 48/12 = 4 \). Wait, so all three sides are in ratio 4:1. So SSS similarity (all sides proportional) and SAS (since the included angle is equal, and two sides around it are proportional). Wait, let's re-express:

For SAS: the two sides around the equal angle. Let's say the equal angle is between \( HF \) and \( FG \) in \( \triangle HFG \), and between \( KL \) and \( JL \) in \( \triangle KJL \). Then \( \frac{HF}{KL} = 36/9 = 4 \), \( \frac{FG}{JL} = 32/8 = 4 \), so the two sides around the equal angle are in proportion (ratio 4), and the included angle is equal, so SAS similarity. Also, all three sides: \( HF/KL = 4 \), \( FG/JL = 4 \), \( HG/KJ = 48/12 = 4 \), so SSS similarity. Wait, but let's check the answer options. The options are: can be used by both SSS and SAS. Let's verify:

SSS: check all three sides. \( \frac{36}{9} = 4 \), \( \frac{32}{8} = 4 \), \( \frac{48}{12} = 4 \). So all sides are in proportion (ratio 4), so SSS similarity holds.

SAS: the included angle (the angle between the two sides) is equal (marked as equal in the diagram), and the two sides around the angle are in proportion. For example, \( \frac{36}{9} = 4 \) and \( \frac{32}{8} = 4 \), so the sides adjacent to the equal angle are in proportion, so SAS similarity holds. Therefore, both SSS and SAS can be used.

Step2: Confirm SSS and SAS

SSS: all three corresponding sides are in proportion (36/9=4, 32/8=4, 48/12=4), so SSS similarity. SAS: the included angle is equal, and the two sides around the angle are in proportion (36/9=4 and 32/8=4), so SAS similarity. Thus, the given sides and angles can be used to show similarity by both 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. (The last option: "The given sides and angles can be used to show similarity by both the SSS and SAS similarity theorems.")