Question

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

Explanation:

To determine if two triangles are similar, we use similarity rules. The SSS (Side - Side - Side) rule requires the ratios of all three pairs of corresponding sides to be equal. The SAS (Side - Angle - Side) rule requires the ratio of two pairs of corresponding sides to be equal and the included angles to be congruent.

For triangle \(STU\) and triangle \(HIJ\):

• Check the ratio of sides: \(\frac{72}{61}

eq\frac{40}{32}\) (since \(\frac{72}{61}\approx1.18\) and \(\frac{40}{32} = 1.25\)). So SSS and SAS rules do not apply as the side - ratios are not equal.

• The AA (Angle - Angle) rule requires two pairs of corresponding angles to be congruent. But we are only given one pair of congruent angles (\(48^{\circ}\)) and no information about the other angles. Also, the side - length ratios (which are needed for SAS if we were to assume the \(48^{\circ}\) angle is the included angle) are not equal.

Answer:

none of the above; the triangles cannot be proven similar