Question

which rule explains why these scalene triangles are similar?
r 36 s
97°
e
97°
33 38
54
f g
t
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Recall Similarity Rules

SSS (Side - Side - Side) similarity requires all three pairs of corresponding sides to be in proportion. SAS (Side - Angle - Side) similarity requires two pairs of corresponding sides to be in proportion and the included angle to be equal. AA (Angle - Angle) similarity requires two pairs of corresponding angles to be equal.

Step2: Check Sides and Angles

First, check the angles: both triangles have a \(97^{\circ}\) angle. Now check the sides around the \(97^{\circ}\) angle. In \(\triangle RST\), the sides around \(\angle S\) ( \(97^{\circ}\)) are \(36\) and \(54\). In \(\triangle EFG\), the sides around \(\angle E\) ( \(97^{\circ}\)) are \(33\) and \(38\). Calculate the ratios: \(\frac{36}{33}=\frac{12}{11}\approx1.09\) and \(\frac{54}{38}=\frac{27}{19}\approx1.42\). These ratios are not equal, so the sides around the equal angle are not in proportion. Also, we only know one pair of equal angles, so AA is not satisfied (needs two pairs). And for SSS, we don't have information about the third side of each triangle to check all three ratios. So the triangles cannot be proven similar using SSS, SAS, or AA.

Answer:

none of the above; the triangles cannot be proven similar