Question
Consider the two triangles shown.
Triangles F H G and L K J are shown. Angles H F G and K L J are congruent. The length of side F G is 32 and the length of side J L is 8. The length of side H G is 48 and the length of side K J is 12. The length of side H F is 36 and the length of side K L is 9.
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.
Triangles F H G and L K J are shown. Angles H F G and K L J are congruent. The length of side F G is 32 and the length of side J L is 8. The length of side H G is 48 and the length of side K J is 12. The length of side H F is 36 and the length of side K L is 9.
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.
Answers
Answered by
GPT-4o mini
To determine whether the triangles FHG and LJK are similar, we can analyze both the given sides and angles.
**Given:**
- Angles \( \angle HFG \) and \( \angle KLJ \) are congruent.
- Side lengths:
- \( FG = 32 \), \( JL = 8 \)
- \( HG = 48 \), \( KJ = 12 \)
- \( HF = 36 \), \( KL = 9 \)
**Step 1: Check angle congruence.**
- We know \( \angle HFG \cong \angle KLJ \).
**Step 2: Check side ratios for SSS similarity.**
- To use the SSS similarity theorem, we must check if the ratios of the corresponding sides are equal:
\[
\frac{FG}{JL} = \frac{32}{8} = 4
\]
\[
\frac{HG}{KJ} = \frac{48}{12} = 4
\]
\[
\frac{HF}{KL} = \frac{36}{9} = 4
\]
The ratios of all corresponding sides are equal (all equal to 4), so we can conclude that the triangles are similar by the SSS similarity theorem.
**Step 3: Check side ratios for SAS similarity.**
- To use the SAS similarity theorem, we can also use one pair of corresponding sides and the included angle:
Using sides \( HF \) and \( KL \) which are proportional to one another:
- \( \frac{HF}{KL} = \frac{36}{9} = 4 \)
- We also have the angle \( \angle HFG \) and \( \angle KLJ \) which are congruent.
Since \( \frac{HF}{KL} = 4 \) and the included angles are equal, we can also conclude similarity by SAS.
**Conclusion:**
Since the triangles have been shown to be similar by both the SSS and SAS similarity theorems, the correct statement is:
**The given sides and angles can be used to show similarity by both the SSS and SAS similarity theorems.**
**Given:**
- Angles \( \angle HFG \) and \( \angle KLJ \) are congruent.
- Side lengths:
- \( FG = 32 \), \( JL = 8 \)
- \( HG = 48 \), \( KJ = 12 \)
- \( HF = 36 \), \( KL = 9 \)
**Step 1: Check angle congruence.**
- We know \( \angle HFG \cong \angle KLJ \).
**Step 2: Check side ratios for SSS similarity.**
- To use the SSS similarity theorem, we must check if the ratios of the corresponding sides are equal:
\[
\frac{FG}{JL} = \frac{32}{8} = 4
\]
\[
\frac{HG}{KJ} = \frac{48}{12} = 4
\]
\[
\frac{HF}{KL} = \frac{36}{9} = 4
\]
The ratios of all corresponding sides are equal (all equal to 4), so we can conclude that the triangles are similar by the SSS similarity theorem.
**Step 3: Check side ratios for SAS similarity.**
- To use the SAS similarity theorem, we can also use one pair of corresponding sides and the included angle:
Using sides \( HF \) and \( KL \) which are proportional to one another:
- \( \frac{HF}{KL} = \frac{36}{9} = 4 \)
- We also have the angle \( \angle HFG \) and \( \angle KLJ \) which are congruent.
Since \( \frac{HF}{KL} = 4 \) and the included angles are equal, we can also conclude similarity by SAS.
**Conclusion:**
Since the triangles have been shown to be similar by both the SSS and SAS similarity theorems, the correct statement is:
**The given sides and angles can be used to show similarity by both the SSS and SAS similarity theorems.**