Question

which rule explains why these triangles are similar?
f 34 h
44 40
g
t
43
u
49
31
s
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Recall Similarity Rules

To determine triangle similarity, we check SSS (side - side - side) similarity (ratios of corresponding sides equal), SAS (side - angle - side) similarity (two sides in ratio and included angle equal), and AA (angle - angle) similarity (two angles equal). For SSS similarity, we need to find the ratios of corresponding sides.

Step2: Identify Corresponding Sides

First, we order the sides of each triangle from smallest to largest.
For $\triangle FGH$: The side lengths are 34, 40, 44. When ordered: 34, 40, 44.
For $\triangle STU$ (wait, the second triangle is $\triangle UST$? Wait, the vertices are U, S, T. The side lengths are 31, 43, 49. When ordered: 31, 43, 49.
Now, we check the ratios of corresponding sides. Let's take the smallest side of the first triangle (34) and the smallest side of the second triangle (31): $\frac{34}{31}\approx1.097$.
Next, the middle - length side of the first triangle (40) and the middle - length side of the second triangle (43): $\frac{40}{43}\approx0.930$.
The largest side of the first triangle (44) and the largest side of the second triangle (49): $\frac{44}{49}\approx0.898$.
Since the ratios of the corresponding sides are not equal, the SSS similarity condition is not met.
For SAS, we need two sides in proportion and the included angle equal. But we don't have information about the included angles, and since the side ratios are not equal, SAS is also not met.
For AA, we don't have any information about the angles. Since the side ratios are not equal (so SSS fails), we don't have angle information for AA or SAS to hold. So the triangles cannot be proven similar using SSS, SAS, or AA.

Answer:

none of the above; the triangles cannot be proven similar