Question

which rule explains why these scalene triangles are similar?
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Calculate the ratio of the sides

For triangle \(UVW\) and \(DEF\), we have \(\frac{56}{49}=\frac{8}{7}\) and \(\frac{72}{77}=\frac{72}{77}\). Since \(\frac{8}{7}
eq\frac{72}{77}\), SSS (Side - Side - Side) similarity (where \(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\)) does not apply.

Step2: Check SAS similarity

SAS similarity requires \(\frac{a}{a'}=\frac{b}{b'}\) and the included angles equal. We have \(\frac{56}{49}=\frac{8}{7}\), \(\frac{72}{77}=\frac{72}{77}\), and \(\frac{8}{7}
eq\frac{72}{77}\), so SAS (Side - Angle - Side) similarity (where \(\frac{\text{side}_1}{\text{side}_1'}=\frac{\text{side}_2}{\text{side}_2'}\) and the included angles are equal) does not hold.

Step3: Check for other similarity rules

Since we have only two sides given for each triangle and we cannot confirm AA (Angle - Angle, which requires two pairs of equal angles) as we have only one angle (\(90^{\circ}\)) given for each triangle and no information about other angles. Also, from the side - ratio calculations above, the triangles do not satisfy SSS or SAS similarity criteria.

Answer:

none of the above; the triangles cannot be proven similar