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, check the included angles. The angle at \( F \) and the angle at \( L \) are equal (marked as equal in the diagram). Now, find the ratios of the sides around these angles. For triangle \( HFG \) and \( KJL \):

• \( \frac{HF}{KJ} = \frac{36}{9} = 4 \)
• \( \frac{FG}{JL} = \frac{32}{8} = 4 \)

Since the included angles are equal and the sides around them are in proportion, SAS similarity holds.

Step2: Check SSS Similarity

Find the ratio of the third sides:

• \( \frac{HG}{KL} = \frac{48}{12} = 4 \)

Now, check all three ratios:

• \( \frac{HF}{KJ} = 4 \), \( \frac{FG}{JL} = 4 \), \( \frac{HG}{KL} = 4 \)

All three sides are in proportion, so SSS similarity also holds.

Answer:

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