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: Check the ratio of sides

Calculate the ratios of corresponding sides. $\frac{32}{60}=\frac{8}{15}$, $\frac{48}{40}=\frac{6}{5}$. The ratios of sides are not equal, so SSS (Side - Side - Side) similarity is not applicable.

Step2: Check SAS similarity

For SAS (Side - Angle - Side) similarity, we need two sides in proportion and the included angle equal. The angle of \(100^{\circ}\) is given. But as we saw in Step 1, the ratios of the adjacent sides to the \(100^{\circ}\) angle (\(\frac{32}{60}
eq\frac{48}{40}\)) are not equal. So SAS similarity is not applicable.

Step3: Check for AA similarity

Since we have only one angle given for each triangle, and we can't confirm another pair of equal angles (no information about other angles is derived from the given side - angle data in a way that would give AA (Angle - Angle) similarity).

Answer:

none of the above; the triangles cannot be proven similar