Question

which rule explains why these scalene triangles are similar?
triangles tus and gfh (with sides and angles marked: tu side 60, us side 52, angle at u; gf side 52, fh side 56, angle at f. options: sss, sas, aa, none of the above; the triangles cannot be proven similar

Explanation:

To determine triangle similarity via SAS, we need two sides in proportion and the included angle equal. For $\triangle TUS$ and $\triangle GFH$: side $US = 52$, $TU = 60$; side $FG = 52$, $FH = 56$. The ratios are $\frac{US}{FG}=\frac{52}{52} = 1$ and $\frac{TU}{FH}=\frac{60}{56}=\frac{15}{14}
eq1$. Since the side ratios are not equal, the included angles (marked) do not satisfy the proportional sides needed for SAS similarity. SSS needs all three sides proportional (not shown), AA needs two angles equal (only one angle marked, not enough). Thus, similarity can’t be proven with given info.

Answer:

none of the above; the triangles cannot be proven similar