Question

which rule explains why these triangles are similar?
q
75
66
s
57
r
u
44
50
v
38
w
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step 1: Calculate the ratios of corresponding sides

First, we identify the corresponding sides. Let's assume triangle \(QRS\) and triangle \(UVW\) (we need to match the sides by their lengths).
For the sides:

• The ratio of \(QR = 66\) and \(UW = 50\)? Wait, no, let's re - order. Let's list the sides of each triangle in ascending order.

Triangle \(QRS\) sides: \(57\), \(66\), \(75\)
Triangle \(UVW\) sides: \(38\), \(44\), \(50\)

Now, calculate the ratios:
\(\frac{57}{38}=\frac{3\times19}{2\times19}=\frac{3}{2} = 1.5\)
\(\frac{66}{44}=\frac{3\times22}{2\times22}=\frac{3}{2}=1.5\)
\(\frac{75}{50}=\frac{3\times25}{2\times25}=\frac{3}{2}=1.5\)

Step 2: Determine the similarity rule

Since the ratios of all three corresponding sides of the two triangles are equal (\(\frac{57}{38}=\frac{66}{44}=\frac{75}{50}=\frac{3}{2}\)), by the SSS (Side - Side - Side) similarity criterion, if the ratios of the three corresponding sides of two triangles are equal, the triangles are similar.

Answer:

SSS (Side - Side - Side) similarity rule