Question

which rule explains why these scalene triangles are similar? triangles fgh and str (with angles and sides labeled) are shown options: sss, sas, aa, none of the above; the triangles cannot be proven similar

Explanation:

To determine triangle similarity, we check SSS (all sides proportional), SAS (two sides proportional and included angle equal), or AA (two angles equal). For these scalene triangles:

• SSS Check: Calculate side ratios. In $\triangle FGH$, sides are 32, 48, and let's find the third side (using Law of Cosines: $FH^2 = 32^2 + 48^2 - 2(32)(48)\cos(97^\circ)$). In $\triangle STR$, sides are 35, 43, and third side $ST^2 = 35^2 + 43^2 - 2(35)(43)\cos(97^\circ)$. The ratios of corresponding sides (32/35, 48/43, and $FH/ST$) are not equal (e.g., $32/35 \approx 0.914$, $48/43 \approx 1.116$), so SSS fails.
• SAS Check: The included angle for SAS must be between the proportional sides. Here, the $97^\circ$ angle is included, but the sides adjacent to $97^\circ$ are 32 & 48 (in $\triangle FGH$) and 35 & 43 (in $\triangle STR$). The ratios $32/35

eq 48/43$, so SAS fails.

• AA Check: Only one angle ($97^\circ$) is known to be equal; we don’t know if another angle matches, so AA fails.

Thus, no similarity rule applies.

Answer:

none of the above; the triangles cannot be proven similar